KILLED proof of input_88FiFnTChz.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (6) CdtProblem (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxRelTRS (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxRelTRS (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (12) typed CpxTrs (13) OrderProof [LOWER BOUND(ID), 9 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 356 ms] (16) BEST (17) proven lower bound (18) LowerBoundPropagationProof [FINISHED, 0 ms] (19) BOUNDS(n^1, INF) (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 175 ms] (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 58 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 84 ms] (26) typed CpxTrs (27) RewriteLemmaProof [LOWER BOUND(ID), 70 ms] (28) typed CpxTrs (29) RewriteLemmaProof [LOWER BOUND(ID), 1996 ms] (30) BOUNDS(1, INF) (31) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (32) CdtProblem (33) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CpxRelTRS (37) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (38) CpxTRS (39) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CpxWeightedTrs (41) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CpxTypedWeightedTrs (43) CompletionProof [UPPER BOUND(ID), 0 ms] (44) CpxTypedWeightedCompleteTrs (45) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CpxTypedWeightedCompleteTrs (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CpxRNTS (51) CompletionProof [UPPER BOUND(ID), 0 ms] (52) CpxTypedWeightedCompleteTrs (53) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 3 ms] (54) CpxRNTS (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 409 ms] (56) CdtProblem (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 39 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 644 ms] (70) CdtProblem (71) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 592 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 53 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 62 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (120) CdtProblem (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 41 ms] (124) CdtProblem (125) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 66 ms] (132) CdtProblem (133) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 39 ms] (136) CdtProblem (137) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 50 ms] (140) CdtProblem (141) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 120 ms] (144) CdtProblem (145) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 93 ms] (148) CdtProblem (149) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 19 ms] (160) CdtProblem (161) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 382 ms] (164) CdtProblem (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtRhsSimplificationProcessorProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 23 ms] (182) CdtProblem (183) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 821 ms] (194) CdtProblem (195) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 657 ms] (204) CdtProblem (205) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 21 ms] (206) CdtProblem (207) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 321 ms] (210) CdtProblem (211) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 10 ms] (212) CdtProblem (213) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 25 ms] (214) CdtProblem (215) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 1580 ms] (218) CdtProblem (219) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (220) CdtProblem (221) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CdtProblem (223) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 500 ms] (224) CdtProblem (225) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (226) CpxWeightedTrs (227) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (228) CpxTypedWeightedTrs (229) CompletionProof [UPPER BOUND(ID), 0 ms] (230) CpxTypedWeightedCompleteTrs (231) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (232) CpxTypedWeightedCompleteTrs (233) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (234) CpxRNTS (235) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (236) CpxRNTS (237) CompletionProof [UPPER BOUND(ID), 0 ms] (238) CpxTypedWeightedCompleteTrs (239) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (240) CpxRNTS ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a__minus(0, Y) -> 0 a__minus(s(X), s(Y)) -> a__minus(X, Y) a__geq(X, 0) -> true a__geq(0, s(Y)) -> false a__geq(s(X), s(Y)) -> a__geq(X, Y) a__div(0, s(Y)) -> 0 a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) mark(minus(X1, X2)) -> a__minus(X1, X2) mark(geq(X1, X2)) -> a__geq(X1, X2) mark(div(X1, X2)) -> a__div(mark(X1), X2) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(true) -> true mark(false) -> false a__minus(X1, X2) -> minus(X1, X2) a__geq(X1, X2) -> geq(X1, X2) a__div(X1, X2) -> div(X1, X2) a__if(X1, X2, X3) -> if(X1, X2, X3) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a__minus(0', Y) -> 0' a__minus(s(X), s(Y)) -> a__minus(X, Y) a__geq(X, 0') -> true a__geq(0', s(Y)) -> false a__geq(s(X), s(Y)) -> a__geq(X, Y) a__div(0', s(Y)) -> 0' a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0') a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) mark(minus(X1, X2)) -> a__minus(X1, X2) mark(geq(X1, X2)) -> a__geq(X1, X2) mark(div(X1, X2)) -> a__div(mark(X1), X2) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(0') -> 0' mark(s(X)) -> s(mark(X)) mark(true) -> true mark(false) -> false a__minus(X1, X2) -> minus(X1, X2) a__geq(X1, X2) -> geq(X1, X2) a__div(X1, X2) -> div(X1, X2) a__if(X1, X2, X3) -> if(X1, X2, X3) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a__minus(0, Y) -> 0 a__minus(s(X), s(Y)) -> a__minus(X, Y) a__geq(X, 0) -> true a__geq(0, s(Y)) -> false a__geq(s(X), s(Y)) -> a__geq(X, Y) a__div(0, s(Y)) -> 0 a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) mark(minus(X1, X2)) -> a__minus(X1, X2) mark(geq(X1, X2)) -> a__geq(X1, X2) mark(div(X1, X2)) -> a__div(mark(X1), X2) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(0) -> 0 mark(s(X)) -> s(mark(X)) mark(true) -> true mark(false) -> false a__minus(X1, X2) -> minus(X1, X2) a__geq(X1, X2) -> geq(X1, X2) a__div(X1, X2) -> div(X1, X2) a__if(X1, X2, X3) -> if(X1, X2, X3) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(0, z0) -> c A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__MINUS(z0, z1) -> c2 A__GEQ(z0, 0) -> c3 A__GEQ(0, s(z0)) -> c4 A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__GEQ(z0, z1) -> c6 A__DIV(0, s(z0)) -> c7 A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) A__DIV(z0, z1) -> c9 A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) A__IF(z0, z1, z2) -> c12 MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(0) -> c17 MARK(s(z0)) -> c18(MARK(z0)) MARK(true) -> c19 MARK(false) -> c20 S tuples: A__MINUS(0, z0) -> c A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__MINUS(z0, z1) -> c2 A__GEQ(z0, 0) -> c3 A__GEQ(0, s(z0)) -> c4 A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__GEQ(z0, z1) -> c6 A__DIV(0, s(z0)) -> c7 A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) A__DIV(z0, z1) -> c9 A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) A__IF(z0, z1, z2) -> c12 MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(0) -> c17 MARK(s(z0)) -> c18(MARK(z0)) MARK(true) -> c19 MARK(false) -> c20 K tuples:none Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__DIV_2, A__IF_3, MARK_1 Compound Symbols: c, c1_1, c2, c3, c4, c5_1, c6, c7, c8_2, c9, c10_1, c11_1, c12, c13_1, c14_1, c15_2, c16_2, c17, c18_1, c19, c20 ---------------------------------------- (7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__MINUS(0, z0) -> c A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__MINUS(z0, z1) -> c2 A__GEQ(z0, 0) -> c3 A__GEQ(0, s(z0)) -> c4 A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__GEQ(z0, z1) -> c6 A__DIV(0, s(z0)) -> c7 A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) A__DIV(z0, z1) -> c9 A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) A__IF(z0, z1, z2) -> c12 MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(0) -> c17 MARK(s(z0)) -> c18(MARK(z0)) MARK(true) -> c19 MARK(false) -> c20 The (relative) TRS S consists of the following rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Rewrite Strategy: INNERMOST ---------------------------------------- (9) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (10) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__MINUS(0', z0) -> c A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__MINUS(z0, z1) -> c2 A__GEQ(z0, 0') -> c3 A__GEQ(0', s(z0)) -> c4 A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__GEQ(z0, z1) -> c6 A__DIV(0', s(z0)) -> c7 A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), A__GEQ(z0, z1)) A__DIV(z0, z1) -> c9 A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) A__IF(z0, z1, z2) -> c12 MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(0') -> c17 MARK(s(z0)) -> c18(MARK(z0)) MARK(true) -> c19 MARK(false) -> c20 The (relative) TRS S consists of the following rules: a__minus(0', z0) -> 0' a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0') -> true a__geq(0', s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0', s(z0)) -> 0' a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0') a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0') -> 0' mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (12) Obligation: Innermost TRS: Rules: A__MINUS(0', z0) -> c A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__MINUS(z0, z1) -> c2 A__GEQ(z0, 0') -> c3 A__GEQ(0', s(z0)) -> c4 A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__GEQ(z0, z1) -> c6 A__DIV(0', s(z0)) -> c7 A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), A__GEQ(z0, z1)) A__DIV(z0, z1) -> c9 A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) A__IF(z0, z1, z2) -> c12 MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(0') -> c17 MARK(s(z0)) -> c18(MARK(z0)) MARK(true) -> c19 MARK(false) -> c20 a__minus(0', z0) -> 0' a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0') -> true a__geq(0', s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0', s(z0)) -> 0' a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0') a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0') -> 0' mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Types: A__MINUS :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c:c1:c2 0' :: 0':s:minus:div:true:false:geq:if c :: c:c1:c2 s :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c1 :: c:c1:c2 -> c:c1:c2 c2 :: c:c1:c2 A__GEQ :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c3:c4:c5:c6 c3 :: c3:c4:c5:c6 c4 :: c3:c4:c5:c6 c5 :: c3:c4:c5:c6 -> c3:c4:c5:c6 c6 :: c3:c4:c5:c6 A__DIV :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c7:c8:c9 c7 :: c7:c8:c9 c8 :: c10:c11:c12 -> c3:c4:c5:c6 -> c7:c8:c9 A__IF :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c10:c11:c12 a__geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c9 :: c7:c8:c9 true :: 0':s:minus:div:true:false:geq:if c10 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 MARK :: 0':s:minus:div:true:false:geq:if -> c13:c14:c15:c16:c17:c18:c19:c20 false :: 0':s:minus:div:true:false:geq:if c11 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 c12 :: c10:c11:c12 c13 :: c:c1:c2 -> c13:c14:c15:c16:c17:c18:c19:c20 geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c14 :: c3:c4:c5:c6 -> c13:c14:c15:c16:c17:c18:c19:c20 c15 :: c7:c8:c9 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 mark :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c16 :: c10:c11:c12 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c17 :: c13:c14:c15:c16:c17:c18:c19:c20 c18 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c19 :: c13:c14:c15:c16:c17:c18:c19:c20 c20 :: c13:c14:c15:c16:c17:c18:c19:c20 a__minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if hole_c:c1:c21_21 :: c:c1:c2 hole_0':s:minus:div:true:false:geq:if2_21 :: 0':s:minus:div:true:false:geq:if hole_c3:c4:c5:c63_21 :: c3:c4:c5:c6 hole_c7:c8:c94_21 :: c7:c8:c9 hole_c10:c11:c125_21 :: c10:c11:c12 hole_c13:c14:c15:c16:c17:c18:c19:c206_21 :: c13:c14:c15:c16:c17:c18:c19:c20 gen_c:c1:c27_21 :: Nat -> c:c1:c2 gen_0':s:minus:div:true:false:geq:if8_21 :: Nat -> 0':s:minus:div:true:false:geq:if gen_c3:c4:c5:c69_21 :: Nat -> c3:c4:c5:c6 gen_c13:c14:c15:c16:c17:c18:c19:c2010_21 :: Nat -> c13:c14:c15:c16:c17:c18:c19:c20 ---------------------------------------- (13) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: A__MINUS, A__GEQ, a__geq, MARK, mark, a__minus They will be analysed ascendingly in the following order: A__MINUS < MARK A__GEQ < MARK a__geq < MARK a__geq < mark mark < MARK a__minus < mark ---------------------------------------- (14) Obligation: Innermost TRS: Rules: A__MINUS(0', z0) -> c A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__MINUS(z0, z1) -> c2 A__GEQ(z0, 0') -> c3 A__GEQ(0', s(z0)) -> c4 A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__GEQ(z0, z1) -> c6 A__DIV(0', s(z0)) -> c7 A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), A__GEQ(z0, z1)) A__DIV(z0, z1) -> c9 A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) A__IF(z0, z1, z2) -> c12 MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(0') -> c17 MARK(s(z0)) -> c18(MARK(z0)) MARK(true) -> c19 MARK(false) -> c20 a__minus(0', z0) -> 0' a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0') -> true a__geq(0', s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0', s(z0)) -> 0' a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0') a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0') -> 0' mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Types: A__MINUS :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c:c1:c2 0' :: 0':s:minus:div:true:false:geq:if c :: c:c1:c2 s :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c1 :: c:c1:c2 -> c:c1:c2 c2 :: c:c1:c2 A__GEQ :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c3:c4:c5:c6 c3 :: c3:c4:c5:c6 c4 :: c3:c4:c5:c6 c5 :: c3:c4:c5:c6 -> c3:c4:c5:c6 c6 :: c3:c4:c5:c6 A__DIV :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c7:c8:c9 c7 :: c7:c8:c9 c8 :: c10:c11:c12 -> c3:c4:c5:c6 -> c7:c8:c9 A__IF :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c10:c11:c12 a__geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c9 :: c7:c8:c9 true :: 0':s:minus:div:true:false:geq:if c10 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 MARK :: 0':s:minus:div:true:false:geq:if -> c13:c14:c15:c16:c17:c18:c19:c20 false :: 0':s:minus:div:true:false:geq:if c11 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 c12 :: c10:c11:c12 c13 :: c:c1:c2 -> c13:c14:c15:c16:c17:c18:c19:c20 geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c14 :: c3:c4:c5:c6 -> c13:c14:c15:c16:c17:c18:c19:c20 c15 :: c7:c8:c9 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 mark :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c16 :: c10:c11:c12 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c17 :: c13:c14:c15:c16:c17:c18:c19:c20 c18 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c19 :: c13:c14:c15:c16:c17:c18:c19:c20 c20 :: c13:c14:c15:c16:c17:c18:c19:c20 a__minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if hole_c:c1:c21_21 :: c:c1:c2 hole_0':s:minus:div:true:false:geq:if2_21 :: 0':s:minus:div:true:false:geq:if hole_c3:c4:c5:c63_21 :: c3:c4:c5:c6 hole_c7:c8:c94_21 :: c7:c8:c9 hole_c10:c11:c125_21 :: c10:c11:c12 hole_c13:c14:c15:c16:c17:c18:c19:c206_21 :: c13:c14:c15:c16:c17:c18:c19:c20 gen_c:c1:c27_21 :: Nat -> c:c1:c2 gen_0':s:minus:div:true:false:geq:if8_21 :: Nat -> 0':s:minus:div:true:false:geq:if gen_c3:c4:c5:c69_21 :: Nat -> c3:c4:c5:c6 gen_c13:c14:c15:c16:c17:c18:c19:c2010_21 :: Nat -> c13:c14:c15:c16:c17:c18:c19:c20 Generator Equations: gen_c:c1:c27_21(0) <=> c gen_c:c1:c27_21(+(x, 1)) <=> c1(gen_c:c1:c27_21(x)) gen_0':s:minus:div:true:false:geq:if8_21(0) <=> 0' gen_0':s:minus:div:true:false:geq:if8_21(+(x, 1)) <=> s(gen_0':s:minus:div:true:false:geq:if8_21(x)) gen_c3:c4:c5:c69_21(0) <=> c3 gen_c3:c4:c5:c69_21(+(x, 1)) <=> c5(gen_c3:c4:c5:c69_21(x)) gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(0) <=> c13(c) gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(+(x, 1)) <=> c15(c7, gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(x)) The following defined symbols remain to be analysed: A__MINUS, A__GEQ, a__geq, MARK, mark, a__minus They will be analysed ascendingly in the following order: A__MINUS < MARK A__GEQ < MARK a__geq < MARK a__geq < mark mark < MARK a__minus < mark ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: A__MINUS(gen_0':s:minus:div:true:false:geq:if8_21(n12_21), gen_0':s:minus:div:true:false:geq:if8_21(n12_21)) -> gen_c:c1:c27_21(n12_21), rt in Omega(1 + n12_21) Induction Base: A__MINUS(gen_0':s:minus:div:true:false:geq:if8_21(0), gen_0':s:minus:div:true:false:geq:if8_21(0)) ->_R^Omega(1) c Induction Step: A__MINUS(gen_0':s:minus:div:true:false:geq:if8_21(+(n12_21, 1)), gen_0':s:minus:div:true:false:geq:if8_21(+(n12_21, 1))) ->_R^Omega(1) c1(A__MINUS(gen_0':s:minus:div:true:false:geq:if8_21(n12_21), gen_0':s:minus:div:true:false:geq:if8_21(n12_21))) ->_IH c1(gen_c:c1:c27_21(c13_21)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (16) Complex Obligation (BEST) ---------------------------------------- (17) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: A__MINUS(0', z0) -> c A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__MINUS(z0, z1) -> c2 A__GEQ(z0, 0') -> c3 A__GEQ(0', s(z0)) -> c4 A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__GEQ(z0, z1) -> c6 A__DIV(0', s(z0)) -> c7 A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), A__GEQ(z0, z1)) A__DIV(z0, z1) -> c9 A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) A__IF(z0, z1, z2) -> c12 MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(0') -> c17 MARK(s(z0)) -> c18(MARK(z0)) MARK(true) -> c19 MARK(false) -> c20 a__minus(0', z0) -> 0' a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0') -> true a__geq(0', s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0', s(z0)) -> 0' a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0') a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0') -> 0' mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Types: A__MINUS :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c:c1:c2 0' :: 0':s:minus:div:true:false:geq:if c :: c:c1:c2 s :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c1 :: c:c1:c2 -> c:c1:c2 c2 :: c:c1:c2 A__GEQ :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c3:c4:c5:c6 c3 :: c3:c4:c5:c6 c4 :: c3:c4:c5:c6 c5 :: c3:c4:c5:c6 -> c3:c4:c5:c6 c6 :: c3:c4:c5:c6 A__DIV :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c7:c8:c9 c7 :: c7:c8:c9 c8 :: c10:c11:c12 -> c3:c4:c5:c6 -> c7:c8:c9 A__IF :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c10:c11:c12 a__geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c9 :: c7:c8:c9 true :: 0':s:minus:div:true:false:geq:if c10 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 MARK :: 0':s:minus:div:true:false:geq:if -> c13:c14:c15:c16:c17:c18:c19:c20 false :: 0':s:minus:div:true:false:geq:if c11 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 c12 :: c10:c11:c12 c13 :: c:c1:c2 -> c13:c14:c15:c16:c17:c18:c19:c20 geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c14 :: c3:c4:c5:c6 -> c13:c14:c15:c16:c17:c18:c19:c20 c15 :: c7:c8:c9 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 mark :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c16 :: c10:c11:c12 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c17 :: c13:c14:c15:c16:c17:c18:c19:c20 c18 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c19 :: c13:c14:c15:c16:c17:c18:c19:c20 c20 :: c13:c14:c15:c16:c17:c18:c19:c20 a__minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if hole_c:c1:c21_21 :: c:c1:c2 hole_0':s:minus:div:true:false:geq:if2_21 :: 0':s:minus:div:true:false:geq:if hole_c3:c4:c5:c63_21 :: c3:c4:c5:c6 hole_c7:c8:c94_21 :: c7:c8:c9 hole_c10:c11:c125_21 :: c10:c11:c12 hole_c13:c14:c15:c16:c17:c18:c19:c206_21 :: c13:c14:c15:c16:c17:c18:c19:c20 gen_c:c1:c27_21 :: Nat -> c:c1:c2 gen_0':s:minus:div:true:false:geq:if8_21 :: Nat -> 0':s:minus:div:true:false:geq:if gen_c3:c4:c5:c69_21 :: Nat -> c3:c4:c5:c6 gen_c13:c14:c15:c16:c17:c18:c19:c2010_21 :: Nat -> c13:c14:c15:c16:c17:c18:c19:c20 Generator Equations: gen_c:c1:c27_21(0) <=> c gen_c:c1:c27_21(+(x, 1)) <=> c1(gen_c:c1:c27_21(x)) gen_0':s:minus:div:true:false:geq:if8_21(0) <=> 0' gen_0':s:minus:div:true:false:geq:if8_21(+(x, 1)) <=> s(gen_0':s:minus:div:true:false:geq:if8_21(x)) gen_c3:c4:c5:c69_21(0) <=> c3 gen_c3:c4:c5:c69_21(+(x, 1)) <=> c5(gen_c3:c4:c5:c69_21(x)) gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(0) <=> c13(c) gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(+(x, 1)) <=> c15(c7, gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(x)) The following defined symbols remain to be analysed: A__MINUS, A__GEQ, a__geq, MARK, mark, a__minus They will be analysed ascendingly in the following order: A__MINUS < MARK A__GEQ < MARK a__geq < MARK a__geq < mark mark < MARK a__minus < mark ---------------------------------------- (18) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (19) BOUNDS(n^1, INF) ---------------------------------------- (20) Obligation: Innermost TRS: Rules: A__MINUS(0', z0) -> c A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__MINUS(z0, z1) -> c2 A__GEQ(z0, 0') -> c3 A__GEQ(0', s(z0)) -> c4 A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__GEQ(z0, z1) -> c6 A__DIV(0', s(z0)) -> c7 A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), A__GEQ(z0, z1)) A__DIV(z0, z1) -> c9 A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) A__IF(z0, z1, z2) -> c12 MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(0') -> c17 MARK(s(z0)) -> c18(MARK(z0)) MARK(true) -> c19 MARK(false) -> c20 a__minus(0', z0) -> 0' a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0') -> true a__geq(0', s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0', s(z0)) -> 0' a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0') a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0') -> 0' mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Types: A__MINUS :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c:c1:c2 0' :: 0':s:minus:div:true:false:geq:if c :: c:c1:c2 s :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c1 :: c:c1:c2 -> c:c1:c2 c2 :: c:c1:c2 A__GEQ :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c3:c4:c5:c6 c3 :: c3:c4:c5:c6 c4 :: c3:c4:c5:c6 c5 :: c3:c4:c5:c6 -> c3:c4:c5:c6 c6 :: c3:c4:c5:c6 A__DIV :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c7:c8:c9 c7 :: c7:c8:c9 c8 :: c10:c11:c12 -> c3:c4:c5:c6 -> c7:c8:c9 A__IF :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c10:c11:c12 a__geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c9 :: c7:c8:c9 true :: 0':s:minus:div:true:false:geq:if c10 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 MARK :: 0':s:minus:div:true:false:geq:if -> c13:c14:c15:c16:c17:c18:c19:c20 false :: 0':s:minus:div:true:false:geq:if c11 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 c12 :: c10:c11:c12 c13 :: c:c1:c2 -> c13:c14:c15:c16:c17:c18:c19:c20 geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c14 :: c3:c4:c5:c6 -> c13:c14:c15:c16:c17:c18:c19:c20 c15 :: c7:c8:c9 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 mark :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c16 :: c10:c11:c12 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c17 :: c13:c14:c15:c16:c17:c18:c19:c20 c18 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c19 :: c13:c14:c15:c16:c17:c18:c19:c20 c20 :: c13:c14:c15:c16:c17:c18:c19:c20 a__minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if hole_c:c1:c21_21 :: c:c1:c2 hole_0':s:minus:div:true:false:geq:if2_21 :: 0':s:minus:div:true:false:geq:if hole_c3:c4:c5:c63_21 :: c3:c4:c5:c6 hole_c7:c8:c94_21 :: c7:c8:c9 hole_c10:c11:c125_21 :: c10:c11:c12 hole_c13:c14:c15:c16:c17:c18:c19:c206_21 :: c13:c14:c15:c16:c17:c18:c19:c20 gen_c:c1:c27_21 :: Nat -> c:c1:c2 gen_0':s:minus:div:true:false:geq:if8_21 :: Nat -> 0':s:minus:div:true:false:geq:if gen_c3:c4:c5:c69_21 :: Nat -> c3:c4:c5:c6 gen_c13:c14:c15:c16:c17:c18:c19:c2010_21 :: Nat -> c13:c14:c15:c16:c17:c18:c19:c20 Lemmas: A__MINUS(gen_0':s:minus:div:true:false:geq:if8_21(n12_21), gen_0':s:minus:div:true:false:geq:if8_21(n12_21)) -> gen_c:c1:c27_21(n12_21), rt in Omega(1 + n12_21) Generator Equations: gen_c:c1:c27_21(0) <=> c gen_c:c1:c27_21(+(x, 1)) <=> c1(gen_c:c1:c27_21(x)) gen_0':s:minus:div:true:false:geq:if8_21(0) <=> 0' gen_0':s:minus:div:true:false:geq:if8_21(+(x, 1)) <=> s(gen_0':s:minus:div:true:false:geq:if8_21(x)) gen_c3:c4:c5:c69_21(0) <=> c3 gen_c3:c4:c5:c69_21(+(x, 1)) <=> c5(gen_c3:c4:c5:c69_21(x)) gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(0) <=> c13(c) gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(+(x, 1)) <=> c15(c7, gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(x)) The following defined symbols remain to be analysed: A__GEQ, a__geq, MARK, mark, a__minus They will be analysed ascendingly in the following order: A__GEQ < MARK a__geq < MARK a__geq < mark mark < MARK a__minus < mark ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: A__GEQ(gen_0':s:minus:div:true:false:geq:if8_21(n713_21), gen_0':s:minus:div:true:false:geq:if8_21(n713_21)) -> gen_c3:c4:c5:c69_21(n713_21), rt in Omega(1 + n713_21) Induction Base: A__GEQ(gen_0':s:minus:div:true:false:geq:if8_21(0), gen_0':s:minus:div:true:false:geq:if8_21(0)) ->_R^Omega(1) c3 Induction Step: A__GEQ(gen_0':s:minus:div:true:false:geq:if8_21(+(n713_21, 1)), gen_0':s:minus:div:true:false:geq:if8_21(+(n713_21, 1))) ->_R^Omega(1) c5(A__GEQ(gen_0':s:minus:div:true:false:geq:if8_21(n713_21), gen_0':s:minus:div:true:false:geq:if8_21(n713_21))) ->_IH c5(gen_c3:c4:c5:c69_21(c714_21)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (22) Obligation: Innermost TRS: Rules: A__MINUS(0', z0) -> c A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__MINUS(z0, z1) -> c2 A__GEQ(z0, 0') -> c3 A__GEQ(0', s(z0)) -> c4 A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__GEQ(z0, z1) -> c6 A__DIV(0', s(z0)) -> c7 A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), A__GEQ(z0, z1)) A__DIV(z0, z1) -> c9 A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) A__IF(z0, z1, z2) -> c12 MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(0') -> c17 MARK(s(z0)) -> c18(MARK(z0)) MARK(true) -> c19 MARK(false) -> c20 a__minus(0', z0) -> 0' a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0') -> true a__geq(0', s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0', s(z0)) -> 0' a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0') a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0') -> 0' mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Types: A__MINUS :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c:c1:c2 0' :: 0':s:minus:div:true:false:geq:if c :: c:c1:c2 s :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c1 :: c:c1:c2 -> c:c1:c2 c2 :: c:c1:c2 A__GEQ :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c3:c4:c5:c6 c3 :: c3:c4:c5:c6 c4 :: c3:c4:c5:c6 c5 :: c3:c4:c5:c6 -> c3:c4:c5:c6 c6 :: c3:c4:c5:c6 A__DIV :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c7:c8:c9 c7 :: c7:c8:c9 c8 :: c10:c11:c12 -> c3:c4:c5:c6 -> c7:c8:c9 A__IF :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c10:c11:c12 a__geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c9 :: c7:c8:c9 true :: 0':s:minus:div:true:false:geq:if c10 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 MARK :: 0':s:minus:div:true:false:geq:if -> c13:c14:c15:c16:c17:c18:c19:c20 false :: 0':s:minus:div:true:false:geq:if c11 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 c12 :: c10:c11:c12 c13 :: c:c1:c2 -> c13:c14:c15:c16:c17:c18:c19:c20 geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c14 :: c3:c4:c5:c6 -> c13:c14:c15:c16:c17:c18:c19:c20 c15 :: c7:c8:c9 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 mark :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c16 :: c10:c11:c12 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c17 :: c13:c14:c15:c16:c17:c18:c19:c20 c18 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c19 :: c13:c14:c15:c16:c17:c18:c19:c20 c20 :: c13:c14:c15:c16:c17:c18:c19:c20 a__minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if hole_c:c1:c21_21 :: c:c1:c2 hole_0':s:minus:div:true:false:geq:if2_21 :: 0':s:minus:div:true:false:geq:if hole_c3:c4:c5:c63_21 :: c3:c4:c5:c6 hole_c7:c8:c94_21 :: c7:c8:c9 hole_c10:c11:c125_21 :: c10:c11:c12 hole_c13:c14:c15:c16:c17:c18:c19:c206_21 :: c13:c14:c15:c16:c17:c18:c19:c20 gen_c:c1:c27_21 :: Nat -> c:c1:c2 gen_0':s:minus:div:true:false:geq:if8_21 :: Nat -> 0':s:minus:div:true:false:geq:if gen_c3:c4:c5:c69_21 :: Nat -> c3:c4:c5:c6 gen_c13:c14:c15:c16:c17:c18:c19:c2010_21 :: Nat -> c13:c14:c15:c16:c17:c18:c19:c20 Lemmas: A__MINUS(gen_0':s:minus:div:true:false:geq:if8_21(n12_21), gen_0':s:minus:div:true:false:geq:if8_21(n12_21)) -> gen_c:c1:c27_21(n12_21), rt in Omega(1 + n12_21) A__GEQ(gen_0':s:minus:div:true:false:geq:if8_21(n713_21), gen_0':s:minus:div:true:false:geq:if8_21(n713_21)) -> gen_c3:c4:c5:c69_21(n713_21), rt in Omega(1 + n713_21) Generator Equations: gen_c:c1:c27_21(0) <=> c gen_c:c1:c27_21(+(x, 1)) <=> c1(gen_c:c1:c27_21(x)) gen_0':s:minus:div:true:false:geq:if8_21(0) <=> 0' gen_0':s:minus:div:true:false:geq:if8_21(+(x, 1)) <=> s(gen_0':s:minus:div:true:false:geq:if8_21(x)) gen_c3:c4:c5:c69_21(0) <=> c3 gen_c3:c4:c5:c69_21(+(x, 1)) <=> c5(gen_c3:c4:c5:c69_21(x)) gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(0) <=> c13(c) gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(+(x, 1)) <=> c15(c7, gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(x)) The following defined symbols remain to be analysed: a__geq, MARK, mark, a__minus They will be analysed ascendingly in the following order: a__geq < MARK a__geq < mark mark < MARK a__minus < mark ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: a__geq(gen_0':s:minus:div:true:false:geq:if8_21(n1753_21), gen_0':s:minus:div:true:false:geq:if8_21(n1753_21)) -> true, rt in Omega(0) Induction Base: a__geq(gen_0':s:minus:div:true:false:geq:if8_21(0), gen_0':s:minus:div:true:false:geq:if8_21(0)) ->_R^Omega(0) true Induction Step: a__geq(gen_0':s:minus:div:true:false:geq:if8_21(+(n1753_21, 1)), gen_0':s:minus:div:true:false:geq:if8_21(+(n1753_21, 1))) ->_R^Omega(0) a__geq(gen_0':s:minus:div:true:false:geq:if8_21(n1753_21), gen_0':s:minus:div:true:false:geq:if8_21(n1753_21)) ->_IH true We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (24) Obligation: Innermost TRS: Rules: A__MINUS(0', z0) -> c A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__MINUS(z0, z1) -> c2 A__GEQ(z0, 0') -> c3 A__GEQ(0', s(z0)) -> c4 A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__GEQ(z0, z1) -> c6 A__DIV(0', s(z0)) -> c7 A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), A__GEQ(z0, z1)) A__DIV(z0, z1) -> c9 A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) A__IF(z0, z1, z2) -> c12 MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(0') -> c17 MARK(s(z0)) -> c18(MARK(z0)) MARK(true) -> c19 MARK(false) -> c20 a__minus(0', z0) -> 0' a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0') -> true a__geq(0', s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0', s(z0)) -> 0' a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0') a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0') -> 0' mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Types: A__MINUS :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c:c1:c2 0' :: 0':s:minus:div:true:false:geq:if c :: c:c1:c2 s :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c1 :: c:c1:c2 -> c:c1:c2 c2 :: c:c1:c2 A__GEQ :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c3:c4:c5:c6 c3 :: c3:c4:c5:c6 c4 :: c3:c4:c5:c6 c5 :: c3:c4:c5:c6 -> c3:c4:c5:c6 c6 :: c3:c4:c5:c6 A__DIV :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c7:c8:c9 c7 :: c7:c8:c9 c8 :: c10:c11:c12 -> c3:c4:c5:c6 -> c7:c8:c9 A__IF :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c10:c11:c12 a__geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c9 :: c7:c8:c9 true :: 0':s:minus:div:true:false:geq:if c10 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 MARK :: 0':s:minus:div:true:false:geq:if -> c13:c14:c15:c16:c17:c18:c19:c20 false :: 0':s:minus:div:true:false:geq:if c11 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 c12 :: c10:c11:c12 c13 :: c:c1:c2 -> c13:c14:c15:c16:c17:c18:c19:c20 geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c14 :: c3:c4:c5:c6 -> c13:c14:c15:c16:c17:c18:c19:c20 c15 :: c7:c8:c9 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 mark :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c16 :: c10:c11:c12 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c17 :: c13:c14:c15:c16:c17:c18:c19:c20 c18 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c19 :: c13:c14:c15:c16:c17:c18:c19:c20 c20 :: c13:c14:c15:c16:c17:c18:c19:c20 a__minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if hole_c:c1:c21_21 :: c:c1:c2 hole_0':s:minus:div:true:false:geq:if2_21 :: 0':s:minus:div:true:false:geq:if hole_c3:c4:c5:c63_21 :: c3:c4:c5:c6 hole_c7:c8:c94_21 :: c7:c8:c9 hole_c10:c11:c125_21 :: c10:c11:c12 hole_c13:c14:c15:c16:c17:c18:c19:c206_21 :: c13:c14:c15:c16:c17:c18:c19:c20 gen_c:c1:c27_21 :: Nat -> c:c1:c2 gen_0':s:minus:div:true:false:geq:if8_21 :: Nat -> 0':s:minus:div:true:false:geq:if gen_c3:c4:c5:c69_21 :: Nat -> c3:c4:c5:c6 gen_c13:c14:c15:c16:c17:c18:c19:c2010_21 :: Nat -> c13:c14:c15:c16:c17:c18:c19:c20 Lemmas: A__MINUS(gen_0':s:minus:div:true:false:geq:if8_21(n12_21), gen_0':s:minus:div:true:false:geq:if8_21(n12_21)) -> gen_c:c1:c27_21(n12_21), rt in Omega(1 + n12_21) A__GEQ(gen_0':s:minus:div:true:false:geq:if8_21(n713_21), gen_0':s:minus:div:true:false:geq:if8_21(n713_21)) -> gen_c3:c4:c5:c69_21(n713_21), rt in Omega(1 + n713_21) a__geq(gen_0':s:minus:div:true:false:geq:if8_21(n1753_21), gen_0':s:minus:div:true:false:geq:if8_21(n1753_21)) -> true, rt in Omega(0) Generator Equations: gen_c:c1:c27_21(0) <=> c gen_c:c1:c27_21(+(x, 1)) <=> c1(gen_c:c1:c27_21(x)) gen_0':s:minus:div:true:false:geq:if8_21(0) <=> 0' gen_0':s:minus:div:true:false:geq:if8_21(+(x, 1)) <=> s(gen_0':s:minus:div:true:false:geq:if8_21(x)) gen_c3:c4:c5:c69_21(0) <=> c3 gen_c3:c4:c5:c69_21(+(x, 1)) <=> c5(gen_c3:c4:c5:c69_21(x)) gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(0) <=> c13(c) gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(+(x, 1)) <=> c15(c7, gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(x)) The following defined symbols remain to be analysed: a__minus, MARK, mark They will be analysed ascendingly in the following order: mark < MARK a__minus < mark ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: a__minus(gen_0':s:minus:div:true:false:geq:if8_21(n2885_21), gen_0':s:minus:div:true:false:geq:if8_21(n2885_21)) -> gen_0':s:minus:div:true:false:geq:if8_21(0), rt in Omega(0) Induction Base: a__minus(gen_0':s:minus:div:true:false:geq:if8_21(0), gen_0':s:minus:div:true:false:geq:if8_21(0)) ->_R^Omega(0) 0' Induction Step: a__minus(gen_0':s:minus:div:true:false:geq:if8_21(+(n2885_21, 1)), gen_0':s:minus:div:true:false:geq:if8_21(+(n2885_21, 1))) ->_R^Omega(0) a__minus(gen_0':s:minus:div:true:false:geq:if8_21(n2885_21), gen_0':s:minus:div:true:false:geq:if8_21(n2885_21)) ->_IH gen_0':s:minus:div:true:false:geq:if8_21(0) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (26) Obligation: Innermost TRS: Rules: A__MINUS(0', z0) -> c A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__MINUS(z0, z1) -> c2 A__GEQ(z0, 0') -> c3 A__GEQ(0', s(z0)) -> c4 A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__GEQ(z0, z1) -> c6 A__DIV(0', s(z0)) -> c7 A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), A__GEQ(z0, z1)) A__DIV(z0, z1) -> c9 A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) A__IF(z0, z1, z2) -> c12 MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(0') -> c17 MARK(s(z0)) -> c18(MARK(z0)) MARK(true) -> c19 MARK(false) -> c20 a__minus(0', z0) -> 0' a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0') -> true a__geq(0', s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0', s(z0)) -> 0' a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0') a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0') -> 0' mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Types: A__MINUS :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c:c1:c2 0' :: 0':s:minus:div:true:false:geq:if c :: c:c1:c2 s :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c1 :: c:c1:c2 -> c:c1:c2 c2 :: c:c1:c2 A__GEQ :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c3:c4:c5:c6 c3 :: c3:c4:c5:c6 c4 :: c3:c4:c5:c6 c5 :: c3:c4:c5:c6 -> c3:c4:c5:c6 c6 :: c3:c4:c5:c6 A__DIV :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c7:c8:c9 c7 :: c7:c8:c9 c8 :: c10:c11:c12 -> c3:c4:c5:c6 -> c7:c8:c9 A__IF :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c10:c11:c12 a__geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c9 :: c7:c8:c9 true :: 0':s:minus:div:true:false:geq:if c10 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 MARK :: 0':s:minus:div:true:false:geq:if -> c13:c14:c15:c16:c17:c18:c19:c20 false :: 0':s:minus:div:true:false:geq:if c11 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 c12 :: c10:c11:c12 c13 :: c:c1:c2 -> c13:c14:c15:c16:c17:c18:c19:c20 geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c14 :: c3:c4:c5:c6 -> c13:c14:c15:c16:c17:c18:c19:c20 c15 :: c7:c8:c9 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 mark :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c16 :: c10:c11:c12 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c17 :: c13:c14:c15:c16:c17:c18:c19:c20 c18 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c19 :: c13:c14:c15:c16:c17:c18:c19:c20 c20 :: c13:c14:c15:c16:c17:c18:c19:c20 a__minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if hole_c:c1:c21_21 :: c:c1:c2 hole_0':s:minus:div:true:false:geq:if2_21 :: 0':s:minus:div:true:false:geq:if hole_c3:c4:c5:c63_21 :: c3:c4:c5:c6 hole_c7:c8:c94_21 :: c7:c8:c9 hole_c10:c11:c125_21 :: c10:c11:c12 hole_c13:c14:c15:c16:c17:c18:c19:c206_21 :: c13:c14:c15:c16:c17:c18:c19:c20 gen_c:c1:c27_21 :: Nat -> c:c1:c2 gen_0':s:minus:div:true:false:geq:if8_21 :: Nat -> 0':s:minus:div:true:false:geq:if gen_c3:c4:c5:c69_21 :: Nat -> c3:c4:c5:c6 gen_c13:c14:c15:c16:c17:c18:c19:c2010_21 :: Nat -> c13:c14:c15:c16:c17:c18:c19:c20 Lemmas: A__MINUS(gen_0':s:minus:div:true:false:geq:if8_21(n12_21), gen_0':s:minus:div:true:false:geq:if8_21(n12_21)) -> gen_c:c1:c27_21(n12_21), rt in Omega(1 + n12_21) A__GEQ(gen_0':s:minus:div:true:false:geq:if8_21(n713_21), gen_0':s:minus:div:true:false:geq:if8_21(n713_21)) -> gen_c3:c4:c5:c69_21(n713_21), rt in Omega(1 + n713_21) a__geq(gen_0':s:minus:div:true:false:geq:if8_21(n1753_21), gen_0':s:minus:div:true:false:geq:if8_21(n1753_21)) -> true, rt in Omega(0) a__minus(gen_0':s:minus:div:true:false:geq:if8_21(n2885_21), gen_0':s:minus:div:true:false:geq:if8_21(n2885_21)) -> gen_0':s:minus:div:true:false:geq:if8_21(0), rt in Omega(0) Generator Equations: gen_c:c1:c27_21(0) <=> c gen_c:c1:c27_21(+(x, 1)) <=> c1(gen_c:c1:c27_21(x)) gen_0':s:minus:div:true:false:geq:if8_21(0) <=> 0' gen_0':s:minus:div:true:false:geq:if8_21(+(x, 1)) <=> s(gen_0':s:minus:div:true:false:geq:if8_21(x)) gen_c3:c4:c5:c69_21(0) <=> c3 gen_c3:c4:c5:c69_21(+(x, 1)) <=> c5(gen_c3:c4:c5:c69_21(x)) gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(0) <=> c13(c) gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(+(x, 1)) <=> c15(c7, gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(x)) The following defined symbols remain to be analysed: mark, MARK They will be analysed ascendingly in the following order: mark < MARK ---------------------------------------- (27) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: mark(gen_0':s:minus:div:true:false:geq:if8_21(n4105_21)) -> gen_0':s:minus:div:true:false:geq:if8_21(n4105_21), rt in Omega(0) Induction Base: mark(gen_0':s:minus:div:true:false:geq:if8_21(0)) ->_R^Omega(0) 0' Induction Step: mark(gen_0':s:minus:div:true:false:geq:if8_21(+(n4105_21, 1))) ->_R^Omega(0) s(mark(gen_0':s:minus:div:true:false:geq:if8_21(n4105_21))) ->_IH s(gen_0':s:minus:div:true:false:geq:if8_21(c4106_21)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (28) Obligation: Innermost TRS: Rules: A__MINUS(0', z0) -> c A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__MINUS(z0, z1) -> c2 A__GEQ(z0, 0') -> c3 A__GEQ(0', s(z0)) -> c4 A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__GEQ(z0, z1) -> c6 A__DIV(0', s(z0)) -> c7 A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0'), A__GEQ(z0, z1)) A__DIV(z0, z1) -> c9 A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) A__IF(z0, z1, z2) -> c12 MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(0') -> c17 MARK(s(z0)) -> c18(MARK(z0)) MARK(true) -> c19 MARK(false) -> c20 a__minus(0', z0) -> 0' a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0') -> true a__geq(0', s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0', s(z0)) -> 0' a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0') a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0') -> 0' mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Types: A__MINUS :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c:c1:c2 0' :: 0':s:minus:div:true:false:geq:if c :: c:c1:c2 s :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c1 :: c:c1:c2 -> c:c1:c2 c2 :: c:c1:c2 A__GEQ :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c3:c4:c5:c6 c3 :: c3:c4:c5:c6 c4 :: c3:c4:c5:c6 c5 :: c3:c4:c5:c6 -> c3:c4:c5:c6 c6 :: c3:c4:c5:c6 A__DIV :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c7:c8:c9 c7 :: c7:c8:c9 c8 :: c10:c11:c12 -> c3:c4:c5:c6 -> c7:c8:c9 A__IF :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> c10:c11:c12 a__geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c9 :: c7:c8:c9 true :: 0':s:minus:div:true:false:geq:if c10 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 MARK :: 0':s:minus:div:true:false:geq:if -> c13:c14:c15:c16:c17:c18:c19:c20 false :: 0':s:minus:div:true:false:geq:if c11 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c10:c11:c12 c12 :: c10:c11:c12 c13 :: c:c1:c2 -> c13:c14:c15:c16:c17:c18:c19:c20 geq :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c14 :: c3:c4:c5:c6 -> c13:c14:c15:c16:c17:c18:c19:c20 c15 :: c7:c8:c9 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 mark :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if c16 :: c10:c11:c12 -> c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c17 :: c13:c14:c15:c16:c17:c18:c19:c20 c18 :: c13:c14:c15:c16:c17:c18:c19:c20 -> c13:c14:c15:c16:c17:c18:c19:c20 c19 :: c13:c14:c15:c16:c17:c18:c19:c20 c20 :: c13:c14:c15:c16:c17:c18:c19:c20 a__minus :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__div :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if a__if :: 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if -> 0':s:minus:div:true:false:geq:if hole_c:c1:c21_21 :: c:c1:c2 hole_0':s:minus:div:true:false:geq:if2_21 :: 0':s:minus:div:true:false:geq:if hole_c3:c4:c5:c63_21 :: c3:c4:c5:c6 hole_c7:c8:c94_21 :: c7:c8:c9 hole_c10:c11:c125_21 :: c10:c11:c12 hole_c13:c14:c15:c16:c17:c18:c19:c206_21 :: c13:c14:c15:c16:c17:c18:c19:c20 gen_c:c1:c27_21 :: Nat -> c:c1:c2 gen_0':s:minus:div:true:false:geq:if8_21 :: Nat -> 0':s:minus:div:true:false:geq:if gen_c3:c4:c5:c69_21 :: Nat -> c3:c4:c5:c6 gen_c13:c14:c15:c16:c17:c18:c19:c2010_21 :: Nat -> c13:c14:c15:c16:c17:c18:c19:c20 Lemmas: A__MINUS(gen_0':s:minus:div:true:false:geq:if8_21(n12_21), gen_0':s:minus:div:true:false:geq:if8_21(n12_21)) -> gen_c:c1:c27_21(n12_21), rt in Omega(1 + n12_21) A__GEQ(gen_0':s:minus:div:true:false:geq:if8_21(n713_21), gen_0':s:minus:div:true:false:geq:if8_21(n713_21)) -> gen_c3:c4:c5:c69_21(n713_21), rt in Omega(1 + n713_21) a__geq(gen_0':s:minus:div:true:false:geq:if8_21(n1753_21), gen_0':s:minus:div:true:false:geq:if8_21(n1753_21)) -> true, rt in Omega(0) a__minus(gen_0':s:minus:div:true:false:geq:if8_21(n2885_21), gen_0':s:minus:div:true:false:geq:if8_21(n2885_21)) -> gen_0':s:minus:div:true:false:geq:if8_21(0), rt in Omega(0) mark(gen_0':s:minus:div:true:false:geq:if8_21(n4105_21)) -> gen_0':s:minus:div:true:false:geq:if8_21(n4105_21), rt in Omega(0) Generator Equations: gen_c:c1:c27_21(0) <=> c gen_c:c1:c27_21(+(x, 1)) <=> c1(gen_c:c1:c27_21(x)) gen_0':s:minus:div:true:false:geq:if8_21(0) <=> 0' gen_0':s:minus:div:true:false:geq:if8_21(+(x, 1)) <=> s(gen_0':s:minus:div:true:false:geq:if8_21(x)) gen_c3:c4:c5:c69_21(0) <=> c3 gen_c3:c4:c5:c69_21(+(x, 1)) <=> c5(gen_c3:c4:c5:c69_21(x)) gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(0) <=> c13(c) gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(+(x, 1)) <=> c15(c7, gen_c13:c14:c15:c16:c17:c18:c19:c2010_21(x)) The following defined symbols remain to be analysed: MARK ---------------------------------------- (29) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: MARK(gen_0':s:minus:div:true:false:geq:if8_21(+(1, n5855_21))) -> *11_21, rt in Omega(n5855_21) Induction Base: MARK(gen_0':s:minus:div:true:false:geq:if8_21(+(1, 0))) Induction Step: MARK(gen_0':s:minus:div:true:false:geq:if8_21(+(1, +(n5855_21, 1)))) ->_R^Omega(1) c18(MARK(gen_0':s:minus:div:true:false:geq:if8_21(+(1, n5855_21)))) ->_IH c18(*11_21) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (30) BOUNDS(1, INF) ---------------------------------------- (31) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(0, z0) -> c A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__MINUS(z0, z1) -> c2 A__GEQ(z0, 0) -> c3 A__GEQ(0, s(z0)) -> c4 A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__GEQ(z0, z1) -> c6 A__DIV(0, s(z0)) -> c7 A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) A__DIV(z0, z1) -> c9 A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) A__IF(z0, z1, z2) -> c12 MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(0) -> c17 MARK(s(z0)) -> c18(MARK(z0)) MARK(true) -> c19 MARK(false) -> c20 S tuples: A__MINUS(0, z0) -> c A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__MINUS(z0, z1) -> c2 A__GEQ(z0, 0) -> c3 A__GEQ(0, s(z0)) -> c4 A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__GEQ(z0, z1) -> c6 A__DIV(0, s(z0)) -> c7 A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) A__DIV(z0, z1) -> c9 A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) A__IF(z0, z1, z2) -> c12 MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(0) -> c17 MARK(s(z0)) -> c18(MARK(z0)) MARK(true) -> c19 MARK(false) -> c20 K tuples:none Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__DIV_2, A__IF_3, MARK_1 Compound Symbols: c, c1_1, c2, c3, c4, c5_1, c6, c7, c8_2, c9, c10_1, c11_1, c12, c13_1, c14_1, c15_2, c16_2, c17, c18_1, c19, c20 ---------------------------------------- (33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: MARK(false) -> c20 MARK(0) -> c17 A__GEQ(z0, 0) -> c3 A__IF(z0, z1, z2) -> c12 A__MINUS(z0, z1) -> c2 A__MINUS(0, z0) -> c MARK(true) -> c19 A__DIV(0, s(z0)) -> c7 A__DIV(z0, z1) -> c9 A__GEQ(0, s(z0)) -> c4 A__GEQ(z0, z1) -> c6 ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) K tuples:none Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__DIV_2, A__IF_3, MARK_1 Compound Symbols: c1_1, c5_1, c8_2, c10_1, c11_1, c13_1, c14_1, c15_2, c16_2, c18_1 ---------------------------------------- (35) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (36) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) The (relative) TRS S consists of the following rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Rewrite Strategy: INNERMOST ---------------------------------------- (37) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (38) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (39) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (40) 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: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) [1] A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) [1] A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) [1] A__IF(true, z0, z1) -> c10(MARK(z0)) [1] A__IF(false, z0, z1) -> c11(MARK(z1)) [1] MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) [1] MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) [1] MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) [1] MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) [1] MARK(s(z0)) -> c18(MARK(z0)) [1] a__minus(0, z0) -> 0 [0] a__minus(s(z0), s(z1)) -> a__minus(z0, z1) [0] a__minus(z0, z1) -> minus(z0, z1) [0] a__geq(z0, 0) -> true [0] a__geq(0, s(z0)) -> false [0] a__geq(s(z0), s(z1)) -> a__geq(z0, z1) [0] a__geq(z0, z1) -> geq(z0, z1) [0] a__div(0, s(z0)) -> 0 [0] a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) [0] a__div(z0, z1) -> div(z0, z1) [0] a__if(true, z0, z1) -> mark(z0) [0] a__if(false, z0, z1) -> mark(z1) [0] a__if(z0, z1, z2) -> if(z0, z1, z2) [0] mark(minus(z0, z1)) -> a__minus(z0, z1) [0] mark(geq(z0, z1)) -> a__geq(z0, z1) [0] mark(div(z0, z1)) -> a__div(mark(z0), z1) [0] mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) [0] mark(0) -> 0 [0] mark(s(z0)) -> s(mark(z0)) [0] mark(true) -> true [0] mark(false) -> false [0] Rewrite Strategy: INNERMOST ---------------------------------------- (41) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (42) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) [1] A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) [1] A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) [1] A__IF(true, z0, z1) -> c10(MARK(z0)) [1] A__IF(false, z0, z1) -> c11(MARK(z1)) [1] MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) [1] MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) [1] MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) [1] MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) [1] MARK(s(z0)) -> c18(MARK(z0)) [1] a__minus(0, z0) -> 0 [0] a__minus(s(z0), s(z1)) -> a__minus(z0, z1) [0] a__minus(z0, z1) -> minus(z0, z1) [0] a__geq(z0, 0) -> true [0] a__geq(0, s(z0)) -> false [0] a__geq(s(z0), s(z1)) -> a__geq(z0, z1) [0] a__geq(z0, z1) -> geq(z0, z1) [0] a__div(0, s(z0)) -> 0 [0] a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) [0] a__div(z0, z1) -> div(z0, z1) [0] a__if(true, z0, z1) -> mark(z0) [0] a__if(false, z0, z1) -> mark(z1) [0] a__if(z0, z1, z2) -> if(z0, z1, z2) [0] mark(minus(z0, z1)) -> a__minus(z0, z1) [0] mark(geq(z0, z1)) -> a__geq(z0, z1) [0] mark(div(z0, z1)) -> a__div(mark(z0), z1) [0] mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) [0] mark(0) -> 0 [0] mark(s(z0)) -> s(mark(z0)) [0] mark(true) -> true [0] mark(false) -> false [0] The TRS has the following type information: A__MINUS :: s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if -> c1 s :: s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if c1 :: c1 -> c1 A__GEQ :: s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if -> c5 c5 :: c5 -> c5 A__DIV :: s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if -> c8 c8 :: c10:c11 -> c5 -> c8 A__IF :: s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if -> c10:c11 a__geq :: s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if div :: s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if minus :: s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if 0 :: s:minus:div:0:true:false:geq:if true :: s:minus:div:0:true:false:geq:if c10 :: c13:c14:c15:c16:c18 -> c10:c11 MARK :: s:minus:div:0:true:false:geq:if -> c13:c14:c15:c16:c18 false :: s:minus:div:0:true:false:geq:if c11 :: c13:c14:c15:c16:c18 -> c10:c11 c13 :: c1 -> c13:c14:c15:c16:c18 geq :: s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if c14 :: c5 -> c13:c14:c15:c16:c18 c15 :: c8 -> c13:c14:c15:c16:c18 -> c13:c14:c15:c16:c18 mark :: s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if if :: s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if c16 :: c10:c11 -> c13:c14:c15:c16:c18 -> c13:c14:c15:c16:c18 c18 :: c13:c14:c15:c16:c18 -> c13:c14:c15:c16:c18 a__minus :: s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if a__div :: s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if a__if :: s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if -> s:minus:div:0:true:false:geq:if Rewrite Strategy: INNERMOST ---------------------------------------- (43) 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: A__MINUS_2 A__GEQ_2 A__DIV_2 A__IF_3 MARK_1 (c) The following functions are completely defined: a__minus_2 a__geq_2 a__div_2 a__if_3 mark_1 Due to the following rules being added: a__minus(v0, v1) -> null_a__minus [0] a__geq(v0, v1) -> null_a__geq [0] a__div(v0, v1) -> null_a__div [0] a__if(v0, v1, v2) -> null_a__if [0] mark(v0) -> null_mark [0] And the following fresh constants: null_a__minus, null_a__geq, null_a__div, null_a__if, null_mark, const, const1, const2, const3, const4 ---------------------------------------- (44) 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: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) [1] A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) [1] A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) [1] A__IF(true, z0, z1) -> c10(MARK(z0)) [1] A__IF(false, z0, z1) -> c11(MARK(z1)) [1] MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) [1] MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) [1] MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) [1] MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) [1] MARK(s(z0)) -> c18(MARK(z0)) [1] a__minus(0, z0) -> 0 [0] a__minus(s(z0), s(z1)) -> a__minus(z0, z1) [0] a__minus(z0, z1) -> minus(z0, z1) [0] a__geq(z0, 0) -> true [0] a__geq(0, s(z0)) -> false [0] a__geq(s(z0), s(z1)) -> a__geq(z0, z1) [0] a__geq(z0, z1) -> geq(z0, z1) [0] a__div(0, s(z0)) -> 0 [0] a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) [0] a__div(z0, z1) -> div(z0, z1) [0] a__if(true, z0, z1) -> mark(z0) [0] a__if(false, z0, z1) -> mark(z1) [0] a__if(z0, z1, z2) -> if(z0, z1, z2) [0] mark(minus(z0, z1)) -> a__minus(z0, z1) [0] mark(geq(z0, z1)) -> a__geq(z0, z1) [0] mark(div(z0, z1)) -> a__div(mark(z0), z1) [0] mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) [0] mark(0) -> 0 [0] mark(s(z0)) -> s(mark(z0)) [0] mark(true) -> true [0] mark(false) -> false [0] a__minus(v0, v1) -> null_a__minus [0] a__geq(v0, v1) -> null_a__geq [0] a__div(v0, v1) -> null_a__div [0] a__if(v0, v1, v2) -> null_a__if [0] mark(v0) -> null_mark [0] The TRS has the following type information: A__MINUS :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> c1 s :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark c1 :: c1 -> c1 A__GEQ :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> c5 c5 :: c5 -> c5 A__DIV :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> c8 c8 :: c10:c11 -> c5 -> c8 A__IF :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> c10:c11 a__geq :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark div :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark minus :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark 0 :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark true :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark c10 :: c13:c14:c15:c16:c18 -> c10:c11 MARK :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> c13:c14:c15:c16:c18 false :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark c11 :: c13:c14:c15:c16:c18 -> c10:c11 c13 :: c1 -> c13:c14:c15:c16:c18 geq :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark c14 :: c5 -> c13:c14:c15:c16:c18 c15 :: c8 -> c13:c14:c15:c16:c18 -> c13:c14:c15:c16:c18 mark :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark if :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark c16 :: c10:c11 -> c13:c14:c15:c16:c18 -> c13:c14:c15:c16:c18 c18 :: c13:c14:c15:c16:c18 -> c13:c14:c15:c16:c18 a__minus :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark a__div :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark a__if :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_a__minus :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_a__geq :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_a__div :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_a__if :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_mark :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark const :: c1 const1 :: c5 const2 :: c8 const3 :: c10:c11 const4 :: c13:c14:c15:c16:c18 Rewrite Strategy: INNERMOST ---------------------------------------- (45) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (46) 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: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) [1] A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) [1] A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0), A__GEQ(z0, 0)) [1] A__DIV(s(0), s(s(z0'))) -> c8(A__IF(false, s(div(minus(0, s(z0')), s(s(z0')))), 0), A__GEQ(0, s(z0'))) [1] A__DIV(s(s(z0'')), s(s(z1'))) -> c8(A__IF(a__geq(z0'', z1'), s(div(minus(s(z0''), s(z1')), s(s(z1')))), 0), A__GEQ(s(z0''), s(z1'))) [1] A__DIV(s(z0), s(z1)) -> c8(A__IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) [1] A__DIV(s(z0), s(z1)) -> c8(A__IF(null_a__geq, s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) [1] A__IF(true, z0, z1) -> c10(MARK(z0)) [1] A__IF(false, z0, z1) -> c11(MARK(z1)) [1] MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) [1] MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) [1] MARK(div(minus(z01, z1''), z1)) -> c15(A__DIV(a__minus(z01, z1''), z1), MARK(minus(z01, z1''))) [1] MARK(div(geq(z02, z11), z1)) -> c15(A__DIV(a__geq(z02, z11), z1), MARK(geq(z02, z11))) [1] MARK(div(div(z03, z12), z1)) -> c15(A__DIV(a__div(mark(z03), z12), z1), MARK(div(z03, z12))) [1] MARK(div(if(z04, z13, z2'), z1)) -> c15(A__DIV(a__if(mark(z04), z13, z2'), z1), MARK(if(z04, z13, z2'))) [1] MARK(div(0, z1)) -> c15(A__DIV(0, z1), MARK(0)) [1] MARK(div(s(z05), z1)) -> c15(A__DIV(s(mark(z05)), z1), MARK(s(z05))) [1] MARK(div(true, z1)) -> c15(A__DIV(true, z1), MARK(true)) [1] MARK(div(false, z1)) -> c15(A__DIV(false, z1), MARK(false)) [1] MARK(div(z0, z1)) -> c15(A__DIV(null_mark, z1), MARK(z0)) [1] MARK(if(minus(z06, z14), z1, z2)) -> c16(A__IF(a__minus(z06, z14), z1, z2), MARK(minus(z06, z14))) [1] MARK(if(geq(z07, z15), z1, z2)) -> c16(A__IF(a__geq(z07, z15), z1, z2), MARK(geq(z07, z15))) [1] MARK(if(div(z08, z16), z1, z2)) -> c16(A__IF(a__div(mark(z08), z16), z1, z2), MARK(div(z08, z16))) [1] MARK(if(if(z09, z17, z2''), z1, z2)) -> c16(A__IF(a__if(mark(z09), z17, z2''), z1, z2), MARK(if(z09, z17, z2''))) [1] MARK(if(0, z1, z2)) -> c16(A__IF(0, z1, z2), MARK(0)) [1] MARK(if(s(z010), z1, z2)) -> c16(A__IF(s(mark(z010)), z1, z2), MARK(s(z010))) [1] MARK(if(true, z1, z2)) -> c16(A__IF(true, z1, z2), MARK(true)) [1] MARK(if(false, z1, z2)) -> c16(A__IF(false, z1, z2), MARK(false)) [1] MARK(if(z0, z1, z2)) -> c16(A__IF(null_mark, z1, z2), MARK(z0)) [1] MARK(s(z0)) -> c18(MARK(z0)) [1] a__minus(0, z0) -> 0 [0] a__minus(s(z0), s(z1)) -> a__minus(z0, z1) [0] a__minus(z0, z1) -> minus(z0, z1) [0] a__geq(z0, 0) -> true [0] a__geq(0, s(z0)) -> false [0] a__geq(s(z0), s(z1)) -> a__geq(z0, z1) [0] a__geq(z0, z1) -> geq(z0, z1) [0] a__div(0, s(z0)) -> 0 [0] a__div(s(z0), s(0)) -> a__if(true, s(div(minus(z0, 0), s(0))), 0) [0] a__div(s(0), s(s(z011))) -> a__if(false, s(div(minus(0, s(z011)), s(s(z011)))), 0) [0] a__div(s(s(z012)), s(s(z18))) -> a__if(a__geq(z012, z18), s(div(minus(s(z012), s(z18)), s(s(z18)))), 0) [0] a__div(s(z0), s(z1)) -> a__if(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) [0] a__div(s(z0), s(z1)) -> a__if(null_a__geq, s(div(minus(z0, z1), s(z1))), 0) [0] a__div(z0, z1) -> div(z0, z1) [0] a__if(true, z0, z1) -> mark(z0) [0] a__if(false, z0, z1) -> mark(z1) [0] a__if(z0, z1, z2) -> if(z0, z1, z2) [0] mark(minus(z0, z1)) -> a__minus(z0, z1) [0] mark(geq(z0, z1)) -> a__geq(z0, z1) [0] mark(div(minus(z013, z19), z1)) -> a__div(a__minus(z013, z19), z1) [0] mark(div(geq(z014, z110), z1)) -> a__div(a__geq(z014, z110), z1) [0] mark(div(div(z015, z111), z1)) -> a__div(a__div(mark(z015), z111), z1) [0] mark(div(if(z016, z112, z21), z1)) -> a__div(a__if(mark(z016), z112, z21), z1) [0] mark(div(0, z1)) -> a__div(0, z1) [0] mark(div(s(z017), z1)) -> a__div(s(mark(z017)), z1) [0] mark(div(true, z1)) -> a__div(true, z1) [0] mark(div(false, z1)) -> a__div(false, z1) [0] mark(div(z0, z1)) -> a__div(null_mark, z1) [0] mark(if(minus(z018, z113), z1, z2)) -> a__if(a__minus(z018, z113), z1, z2) [0] mark(if(geq(z019, z114), z1, z2)) -> a__if(a__geq(z019, z114), z1, z2) [0] mark(if(div(z020, z115), z1, z2)) -> a__if(a__div(mark(z020), z115), z1, z2) [0] mark(if(if(z021, z116, z22), z1, z2)) -> a__if(a__if(mark(z021), z116, z22), z1, z2) [0] mark(if(0, z1, z2)) -> a__if(0, z1, z2) [0] mark(if(s(z022), z1, z2)) -> a__if(s(mark(z022)), z1, z2) [0] mark(if(true, z1, z2)) -> a__if(true, z1, z2) [0] mark(if(false, z1, z2)) -> a__if(false, z1, z2) [0] mark(if(z0, z1, z2)) -> a__if(null_mark, z1, z2) [0] mark(0) -> 0 [0] mark(s(z0)) -> s(mark(z0)) [0] mark(true) -> true [0] mark(false) -> false [0] a__minus(v0, v1) -> null_a__minus [0] a__geq(v0, v1) -> null_a__geq [0] a__div(v0, v1) -> null_a__div [0] a__if(v0, v1, v2) -> null_a__if [0] mark(v0) -> null_mark [0] The TRS has the following type information: A__MINUS :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> c1 s :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark c1 :: c1 -> c1 A__GEQ :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> c5 c5 :: c5 -> c5 A__DIV :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> c8 c8 :: c10:c11 -> c5 -> c8 A__IF :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> c10:c11 a__geq :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark div :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark minus :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark 0 :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark true :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark c10 :: c13:c14:c15:c16:c18 -> c10:c11 MARK :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> c13:c14:c15:c16:c18 false :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark c11 :: c13:c14:c15:c16:c18 -> c10:c11 c13 :: c1 -> c13:c14:c15:c16:c18 geq :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark c14 :: c5 -> c13:c14:c15:c16:c18 c15 :: c8 -> c13:c14:c15:c16:c18 -> c13:c14:c15:c16:c18 mark :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark if :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark c16 :: c10:c11 -> c13:c14:c15:c16:c18 -> c13:c14:c15:c16:c18 c18 :: c13:c14:c15:c16:c18 -> c13:c14:c15:c16:c18 a__minus :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark a__div :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark a__if :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_a__minus :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_a__geq :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_a__div :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_a__if :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_mark :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark const :: c1 const1 :: c5 const2 :: c8 const3 :: c10:c11 const4 :: c13:c14:c15:c16:c18 Rewrite Strategy: INNERMOST ---------------------------------------- (47) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 false => 1 null_a__minus => 0 null_a__geq => 0 null_a__div => 0 null_a__if => 0 null_mark => 0 const => 0 const1 => 0 const2 => 0 const3 => 0 const4 => 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: A__DIV(z, z') -{ 1 }-> 1 + A__IF(a__geq(z0'', z1'), 1 + (1 + (1 + (1 + z0'') + (1 + z1')) + (1 + (1 + z1'))), 0) + A__GEQ(1 + z0'', 1 + z1') :|: z' = 1 + (1 + z1'), z = 1 + (1 + z0''), z1' >= 0, z0'' >= 0 A__DIV(z, z') -{ 1 }-> 1 + A__IF(2, 1 + (1 + (1 + z0 + 0) + (1 + 0)), 0) + A__GEQ(z0, 0) :|: z = 1 + z0, z' = 1 + 0, z0 >= 0 A__DIV(z, z') -{ 1 }-> 1 + A__IF(1, 1 + (1 + (1 + 0 + (1 + z0')) + (1 + (1 + z0'))), 0) + A__GEQ(0, 1 + z0') :|: z' = 1 + (1 + z0'), z0' >= 0, z = 1 + 0 A__DIV(z, z') -{ 1 }-> 1 + A__IF(0, 1 + (1 + (1 + z0 + z1) + (1 + z1)), 0) + A__GEQ(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 A__DIV(z, z') -{ 1 }-> 1 + A__IF(1 + z0 + z1, 1 + (1 + (1 + z0 + z1) + (1 + z1)), 0) + A__GEQ(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 A__GEQ(z, z') -{ 1 }-> 1 + A__GEQ(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 A__IF(z, z', z'') -{ 1 }-> 1 + MARK(z0) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1 A__IF(z, z', z'') -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1 A__MINUS(z, z') -{ 1 }-> 1 + A__MINUS(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__MINUS(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__GEQ(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__IF(a__minus(z06, z14), z1, z2) + MARK(1 + z06 + z14) :|: z1 >= 0, z06 >= 0, z = 1 + (1 + z06 + z14) + z1 + z2, z14 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__if(mark(z09), z17, z2''), z1, z2) + MARK(1 + z09 + z17 + z2'') :|: z1 >= 0, z17 >= 0, z09 >= 0, z = 1 + (1 + z09 + z17 + z2'') + z1 + z2, z2'' >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__geq(z07, z15), z1, z2) + MARK(1 + z07 + z15) :|: z15 >= 0, z1 >= 0, z07 >= 0, z = 1 + (1 + z07 + z15) + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__div(mark(z08), z16), z1, z2) + MARK(1 + z08 + z16) :|: z08 >= 0, z1 >= 0, z16 >= 0, z2 >= 0, z = 1 + (1 + z08 + z16) + z1 + z2 MARK(z) -{ 1 }-> 1 + A__IF(2, z1, z2) + MARK(2) :|: z1 >= 0, z = 1 + 2 + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(1, z1, z2) + MARK(1) :|: z1 >= 0, z = 1 + 1 + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(0, z1, z2) + MARK(z0) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(0, z1, z2) + MARK(0) :|: z1 >= 0, z = 1 + 0 + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(1 + mark(z010), z1, z2) + MARK(1 + z010) :|: z1 >= 0, z = 1 + (1 + z010) + z1 + z2, z010 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__DIV(a__minus(z01, z1''), z1) + MARK(1 + z01 + z1'') :|: z = 1 + (1 + z01 + z1'') + z1, z1 >= 0, z01 >= 0, z1'' >= 0 MARK(z) -{ 1 }-> 1 + A__DIV(a__if(mark(z04), z13, z2'), z1) + MARK(1 + z04 + z13 + z2') :|: z04 >= 0, z1 >= 0, z = 1 + (1 + z04 + z13 + z2') + z1, z2' >= 0, z13 >= 0 MARK(z) -{ 1 }-> 1 + A__DIV(a__geq(z02, z11), z1) + MARK(1 + z02 + z11) :|: z = 1 + (1 + z02 + z11) + z1, z11 >= 0, z1 >= 0, z02 >= 0 MARK(z) -{ 1 }-> 1 + A__DIV(a__div(mark(z03), z12), z1) + MARK(1 + z03 + z12) :|: z1 >= 0, z12 >= 0, z = 1 + (1 + z03 + z12) + z1, z03 >= 0 MARK(z) -{ 1 }-> 1 + A__DIV(2, z1) + MARK(2) :|: z1 >= 0, z = 1 + 2 + z1 MARK(z) -{ 1 }-> 1 + A__DIV(1, z1) + MARK(1) :|: z1 >= 0, z = 1 + 1 + z1 MARK(z) -{ 1 }-> 1 + A__DIV(0, z1) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__DIV(0, z1) + MARK(0) :|: z1 >= 0, z = 1 + 0 + z1 MARK(z) -{ 1 }-> 1 + A__DIV(1 + mark(z05), z1) + MARK(1 + z05) :|: z1 >= 0, z = 1 + (1 + z05) + z1, z05 >= 0 a__div(z, z') -{ 0 }-> a__if(a__geq(z012, z18), 1 + (1 + (1 + (1 + z012) + (1 + z18)) + (1 + (1 + z18))), 0) :|: z18 >= 0, z' = 1 + (1 + z18), z012 >= 0, z = 1 + (1 + z012) a__div(z, z') -{ 0 }-> a__if(2, 1 + (1 + (1 + z0 + 0) + (1 + 0)), 0) :|: z = 1 + z0, z' = 1 + 0, z0 >= 0 a__div(z, z') -{ 0 }-> a__if(1, 1 + (1 + (1 + 0 + (1 + z011)) + (1 + (1 + z011))), 0) :|: z011 >= 0, z = 1 + 0, z' = 1 + (1 + z011) a__div(z, z') -{ 0 }-> a__if(0, 1 + (1 + (1 + z0 + z1) + (1 + z1)), 0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 a__div(z, z') -{ 0 }-> a__if(1 + z0 + z1, 1 + (1 + (1 + z0 + z1) + (1 + z1)), 0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 a__div(z, z') -{ 0 }-> 0 :|: z0 >= 0, z' = 1 + z0, z = 0 a__div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a__div(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 a__geq(z, z') -{ 0 }-> a__geq(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 a__geq(z, z') -{ 0 }-> 2 :|: z = z0, z0 >= 0, z' = 0 a__geq(z, z') -{ 0 }-> 1 :|: z0 >= 0, z' = 1 + z0, z = 0 a__geq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a__geq(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 a__if(z, z', z'') -{ 0 }-> mark(z0) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1 a__if(z, z', z'') -{ 0 }-> mark(z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1 a__if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 a__if(z, z', z'') -{ 0 }-> 1 + z0 + z1 + z2 :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0 a__minus(z, z') -{ 0 }-> a__minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 a__minus(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0 a__minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a__minus(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 mark(z) -{ 0 }-> a__minus(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> a__if(a__minus(z018, z113), z1, z2) :|: z1 >= 0, z113 >= 0, z = 1 + (1 + z018 + z113) + z1 + z2, z018 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(a__if(mark(z021), z116, z22), z1, z2) :|: z021 >= 0, z1 >= 0, z = 1 + (1 + z021 + z116 + z22) + z1 + z2, z22 >= 0, z2 >= 0, z116 >= 0 mark(z) -{ 0 }-> a__if(a__geq(z019, z114), z1, z2) :|: z019 >= 0, z114 >= 0, z1 >= 0, z = 1 + (1 + z019 + z114) + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(a__div(mark(z020), z115), z1, z2) :|: z1 >= 0, z115 >= 0, z020 >= 0, z2 >= 0, z = 1 + (1 + z020 + z115) + z1 + z2 mark(z) -{ 0 }-> a__if(2, z1, z2) :|: z1 >= 0, z = 1 + 2 + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(1, z1, z2) :|: z1 >= 0, z = 1 + 1 + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(0, z1, z2) :|: z1 >= 0, z = 1 + 0 + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(0, z1, z2) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(1 + mark(z022), z1, z2) :|: z = 1 + (1 + z022) + z1 + z2, z1 >= 0, z022 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__geq(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> a__div(a__minus(z013, z19), z1) :|: z = 1 + (1 + z013 + z19) + z1, z1 >= 0, z19 >= 0, z013 >= 0 mark(z) -{ 0 }-> a__div(a__if(mark(z016), z112, z21), z1) :|: z21 >= 0, z1 >= 0, z = 1 + (1 + z016 + z112 + z21) + z1, z112 >= 0, z016 >= 0 mark(z) -{ 0 }-> a__div(a__geq(z014, z110), z1) :|: z110 >= 0, z1 >= 0, z014 >= 0, z = 1 + (1 + z014 + z110) + z1 mark(z) -{ 0 }-> a__div(a__div(mark(z015), z111), z1) :|: z015 >= 0, z1 >= 0, z = 1 + (1 + z015 + z111) + z1, z111 >= 0 mark(z) -{ 0 }-> a__div(2, z1) :|: z1 >= 0, z = 1 + 2 + z1 mark(z) -{ 0 }-> a__div(1, z1) :|: z1 >= 0, z = 1 + 1 + z1 mark(z) -{ 0 }-> a__div(0, z1) :|: z1 >= 0, z = 1 + 0 + z1 mark(z) -{ 0 }-> a__div(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> a__div(1 + mark(z017), z1) :|: z1 >= 0, z = 1 + (1 + z017) + z1, z017 >= 0 mark(z) -{ 0 }-> 2 :|: z = 2 mark(z) -{ 0 }-> 1 :|: z = 1 mark(z) -{ 0 }-> 0 :|: z = 0 mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 mark(z) -{ 0 }-> 1 + mark(z0) :|: z = 1 + z0, z0 >= 0 ---------------------------------------- (49) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: A__DIV(z, z') -{ 1 }-> 1 + A__IF(a__geq(z - 2, z' - 2), 1 + (1 + (1 + (1 + (z - 2)) + (1 + (z' - 2))) + (1 + (1 + (z' - 2)))), 0) + A__GEQ(1 + (z - 2), 1 + (z' - 2)) :|: z' - 2 >= 0, z - 2 >= 0 A__DIV(z, z') -{ 1 }-> 1 + A__IF(2, 1 + (1 + (1 + (z - 1) + 0) + (1 + 0)), 0) + A__GEQ(z - 1, 0) :|: z' = 1 + 0, z - 1 >= 0 A__DIV(z, z') -{ 1 }-> 1 + A__IF(1, 1 + (1 + (1 + 0 + (1 + (z' - 2))) + (1 + (1 + (z' - 2)))), 0) + A__GEQ(0, 1 + (z' - 2)) :|: z' - 2 >= 0, z = 1 + 0 A__DIV(z, z') -{ 1 }-> 1 + A__IF(0, 1 + (1 + (1 + (z - 1) + (z' - 1)) + (1 + (z' - 1))), 0) + A__GEQ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 A__DIV(z, z') -{ 1 }-> 1 + A__IF(1 + (z - 1) + (z' - 1), 1 + (1 + (1 + (z - 1) + (z' - 1)) + (1 + (z' - 1))), 0) + A__GEQ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 A__GEQ(z, z') -{ 1 }-> 1 + A__GEQ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 A__IF(z, z', z'') -{ 1 }-> 1 + MARK(z') :|: z = 2, z'' >= 0, z' >= 0 A__IF(z, z', z'') -{ 1 }-> 1 + MARK(z'') :|: z'' >= 0, z = 1, z' >= 0 A__MINUS(z, z') -{ 1 }-> 1 + A__MINUS(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 MARK(z) -{ 1 }-> 1 + MARK(z - 1) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__MINUS(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__GEQ(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__IF(a__minus(z06, z14), z1, z2) + MARK(1 + z06 + z14) :|: z1 >= 0, z06 >= 0, z = 1 + (1 + z06 + z14) + z1 + z2, z14 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__if(mark(z09), z17, z2''), z1, z2) + MARK(1 + z09 + z17 + z2'') :|: z1 >= 0, z17 >= 0, z09 >= 0, z = 1 + (1 + z09 + z17 + z2'') + z1 + z2, z2'' >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__geq(z07, z15), z1, z2) + MARK(1 + z07 + z15) :|: z15 >= 0, z1 >= 0, z07 >= 0, z = 1 + (1 + z07 + z15) + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__div(mark(z08), z16), z1, z2) + MARK(1 + z08 + z16) :|: z08 >= 0, z1 >= 0, z16 >= 0, z2 >= 0, z = 1 + (1 + z08 + z16) + z1 + z2 MARK(z) -{ 1 }-> 1 + A__IF(2, z1, z2) + MARK(2) :|: z1 >= 0, z = 1 + 2 + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(1, z1, z2) + MARK(1) :|: z1 >= 0, z = 1 + 1 + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(0, z1, z2) + MARK(z0) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(0, z1, z2) + MARK(0) :|: z1 >= 0, z = 1 + 0 + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(1 + mark(z010), z1, z2) + MARK(1 + z010) :|: z1 >= 0, z = 1 + (1 + z010) + z1 + z2, z010 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__DIV(a__minus(z01, z1''), z1) + MARK(1 + z01 + z1'') :|: z = 1 + (1 + z01 + z1'') + z1, z1 >= 0, z01 >= 0, z1'' >= 0 MARK(z) -{ 1 }-> 1 + A__DIV(a__if(mark(z04), z13, z2'), z1) + MARK(1 + z04 + z13 + z2') :|: z04 >= 0, z1 >= 0, z = 1 + (1 + z04 + z13 + z2') + z1, z2' >= 0, z13 >= 0 MARK(z) -{ 1 }-> 1 + A__DIV(a__geq(z02, z11), z1) + MARK(1 + z02 + z11) :|: z = 1 + (1 + z02 + z11) + z1, z11 >= 0, z1 >= 0, z02 >= 0 MARK(z) -{ 1 }-> 1 + A__DIV(a__div(mark(z03), z12), z1) + MARK(1 + z03 + z12) :|: z1 >= 0, z12 >= 0, z = 1 + (1 + z03 + z12) + z1, z03 >= 0 MARK(z) -{ 1 }-> 1 + A__DIV(2, z - 3) + MARK(2) :|: z - 3 >= 0 MARK(z) -{ 1 }-> 1 + A__DIV(1, z - 2) + MARK(1) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__DIV(0, z1) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__DIV(0, z - 1) + MARK(0) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__DIV(1 + mark(z05), z1) + MARK(1 + z05) :|: z1 >= 0, z = 1 + (1 + z05) + z1, z05 >= 0 a__div(z, z') -{ 0 }-> a__if(a__geq(z - 2, z' - 2), 1 + (1 + (1 + (1 + (z - 2)) + (1 + (z' - 2))) + (1 + (1 + (z' - 2)))), 0) :|: z' - 2 >= 0, z - 2 >= 0 a__div(z, z') -{ 0 }-> a__if(2, 1 + (1 + (1 + (z - 1) + 0) + (1 + 0)), 0) :|: z' = 1 + 0, z - 1 >= 0 a__div(z, z') -{ 0 }-> a__if(1, 1 + (1 + (1 + 0 + (1 + (z' - 2))) + (1 + (1 + (z' - 2)))), 0) :|: z' - 2 >= 0, z = 1 + 0 a__div(z, z') -{ 0 }-> a__if(0, 1 + (1 + (1 + (z - 1) + (z' - 1)) + (1 + (z' - 1))), 0) :|: z' - 1 >= 0, z - 1 >= 0 a__div(z, z') -{ 0 }-> a__if(1 + (z - 1) + (z' - 1), 1 + (1 + (1 + (z - 1) + (z' - 1)) + (1 + (z' - 1))), 0) :|: z' - 1 >= 0, z - 1 >= 0 a__div(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0 a__div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 a__div(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0 a__geq(z, z') -{ 0 }-> a__geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 a__geq(z, z') -{ 0 }-> 2 :|: z >= 0, z' = 0 a__geq(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0 a__geq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 a__geq(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0 a__if(z, z', z'') -{ 0 }-> mark(z') :|: z = 2, z'' >= 0, z' >= 0 a__if(z, z', z'') -{ 0 }-> mark(z'') :|: z'' >= 0, z = 1, z' >= 0 a__if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 a__if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z' >= 0, z >= 0, z'' >= 0 a__minus(z, z') -{ 0 }-> a__minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 a__minus(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0 a__minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 a__minus(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0 mark(z) -{ 0 }-> a__minus(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> a__if(a__minus(z018, z113), z1, z2) :|: z1 >= 0, z113 >= 0, z = 1 + (1 + z018 + z113) + z1 + z2, z018 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(a__if(mark(z021), z116, z22), z1, z2) :|: z021 >= 0, z1 >= 0, z = 1 + (1 + z021 + z116 + z22) + z1 + z2, z22 >= 0, z2 >= 0, z116 >= 0 mark(z) -{ 0 }-> a__if(a__geq(z019, z114), z1, z2) :|: z019 >= 0, z114 >= 0, z1 >= 0, z = 1 + (1 + z019 + z114) + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(a__div(mark(z020), z115), z1, z2) :|: z1 >= 0, z115 >= 0, z020 >= 0, z2 >= 0, z = 1 + (1 + z020 + z115) + z1 + z2 mark(z) -{ 0 }-> a__if(2, z1, z2) :|: z1 >= 0, z = 1 + 2 + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(1, z1, z2) :|: z1 >= 0, z = 1 + 1 + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(0, z1, z2) :|: z1 >= 0, z = 1 + 0 + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(0, z1, z2) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(1 + mark(z022), z1, z2) :|: z = 1 + (1 + z022) + z1 + z2, z1 >= 0, z022 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__geq(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> a__div(a__minus(z013, z19), z1) :|: z = 1 + (1 + z013 + z19) + z1, z1 >= 0, z19 >= 0, z013 >= 0 mark(z) -{ 0 }-> a__div(a__if(mark(z016), z112, z21), z1) :|: z21 >= 0, z1 >= 0, z = 1 + (1 + z016 + z112 + z21) + z1, z112 >= 0, z016 >= 0 mark(z) -{ 0 }-> a__div(a__geq(z014, z110), z1) :|: z110 >= 0, z1 >= 0, z014 >= 0, z = 1 + (1 + z014 + z110) + z1 mark(z) -{ 0 }-> a__div(a__div(mark(z015), z111), z1) :|: z015 >= 0, z1 >= 0, z = 1 + (1 + z015 + z111) + z1, z111 >= 0 mark(z) -{ 0 }-> a__div(2, z - 3) :|: z - 3 >= 0 mark(z) -{ 0 }-> a__div(1, z - 2) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__div(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> a__div(0, z - 1) :|: z - 1 >= 0 mark(z) -{ 0 }-> a__div(1 + mark(z017), z1) :|: z1 >= 0, z = 1 + (1 + z017) + z1, z017 >= 0 mark(z) -{ 0 }-> 2 :|: z = 2 mark(z) -{ 0 }-> 1 :|: z = 1 mark(z) -{ 0 }-> 0 :|: z = 0 mark(z) -{ 0 }-> 0 :|: z >= 0 mark(z) -{ 0 }-> 1 + mark(z - 1) :|: z - 1 >= 0 ---------------------------------------- (51) 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: a__minus(v0, v1) -> null_a__minus [0] a__geq(v0, v1) -> null_a__geq [0] a__div(v0, v1) -> null_a__div [0] a__if(v0, v1, v2) -> null_a__if [0] mark(v0) -> null_mark [0] A__MINUS(v0, v1) -> null_A__MINUS [0] A__GEQ(v0, v1) -> null_A__GEQ [0] A__DIV(v0, v1) -> null_A__DIV [0] A__IF(v0, v1, v2) -> null_A__IF [0] MARK(v0) -> null_MARK [0] And the following fresh constants: null_a__minus, null_a__geq, null_a__div, null_a__if, null_mark, null_A__MINUS, null_A__GEQ, null_A__DIV, null_A__IF, null_MARK ---------------------------------------- (52) 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: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) [1] A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) [1] A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) [1] A__IF(true, z0, z1) -> c10(MARK(z0)) [1] A__IF(false, z0, z1) -> c11(MARK(z1)) [1] MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) [1] MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) [1] MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) [1] MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) [1] MARK(s(z0)) -> c18(MARK(z0)) [1] a__minus(0, z0) -> 0 [0] a__minus(s(z0), s(z1)) -> a__minus(z0, z1) [0] a__minus(z0, z1) -> minus(z0, z1) [0] a__geq(z0, 0) -> true [0] a__geq(0, s(z0)) -> false [0] a__geq(s(z0), s(z1)) -> a__geq(z0, z1) [0] a__geq(z0, z1) -> geq(z0, z1) [0] a__div(0, s(z0)) -> 0 [0] a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) [0] a__div(z0, z1) -> div(z0, z1) [0] a__if(true, z0, z1) -> mark(z0) [0] a__if(false, z0, z1) -> mark(z1) [0] a__if(z0, z1, z2) -> if(z0, z1, z2) [0] mark(minus(z0, z1)) -> a__minus(z0, z1) [0] mark(geq(z0, z1)) -> a__geq(z0, z1) [0] mark(div(z0, z1)) -> a__div(mark(z0), z1) [0] mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) [0] mark(0) -> 0 [0] mark(s(z0)) -> s(mark(z0)) [0] mark(true) -> true [0] mark(false) -> false [0] a__minus(v0, v1) -> null_a__minus [0] a__geq(v0, v1) -> null_a__geq [0] a__div(v0, v1) -> null_a__div [0] a__if(v0, v1, v2) -> null_a__if [0] mark(v0) -> null_mark [0] A__MINUS(v0, v1) -> null_A__MINUS [0] A__GEQ(v0, v1) -> null_A__GEQ [0] A__DIV(v0, v1) -> null_A__DIV [0] A__IF(v0, v1, v2) -> null_A__IF [0] MARK(v0) -> null_MARK [0] The TRS has the following type information: A__MINUS :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> c1:null_A__MINUS s :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark c1 :: c1:null_A__MINUS -> c1:null_A__MINUS A__GEQ :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> c5:null_A__GEQ c5 :: c5:null_A__GEQ -> c5:null_A__GEQ A__DIV :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> c8:null_A__DIV c8 :: c10:c11:null_A__IF -> c5:null_A__GEQ -> c8:null_A__DIV A__IF :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> c10:c11:null_A__IF a__geq :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark div :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark minus :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark 0 :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark true :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark c10 :: c13:c14:c15:c16:c18:null_MARK -> c10:c11:null_A__IF MARK :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> c13:c14:c15:c16:c18:null_MARK false :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark c11 :: c13:c14:c15:c16:c18:null_MARK -> c10:c11:null_A__IF c13 :: c1:null_A__MINUS -> c13:c14:c15:c16:c18:null_MARK geq :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark c14 :: c5:null_A__GEQ -> c13:c14:c15:c16:c18:null_MARK c15 :: c8:null_A__DIV -> c13:c14:c15:c16:c18:null_MARK -> c13:c14:c15:c16:c18:null_MARK mark :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark if :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark c16 :: c10:c11:null_A__IF -> c13:c14:c15:c16:c18:null_MARK -> c13:c14:c15:c16:c18:null_MARK c18 :: c13:c14:c15:c16:c18:null_MARK -> c13:c14:c15:c16:c18:null_MARK a__minus :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark a__div :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark a__if :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark -> s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_a__minus :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_a__geq :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_a__div :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_a__if :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_mark :: s:minus:div:0:true:false:geq:if:null_a__minus:null_a__geq:null_a__div:null_a__if:null_mark null_A__MINUS :: c1:null_A__MINUS null_A__GEQ :: c5:null_A__GEQ null_A__DIV :: c8:null_A__DIV null_A__IF :: c10:c11:null_A__IF null_MARK :: c13:c14:c15:c16:c18:null_MARK Rewrite Strategy: INNERMOST ---------------------------------------- (53) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 false => 1 null_a__minus => 0 null_a__geq => 0 null_a__div => 0 null_a__if => 0 null_mark => 0 null_A__MINUS => 0 null_A__GEQ => 0 null_A__DIV => 0 null_A__IF => 0 null_MARK => 0 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: A__DIV(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 A__DIV(z, z') -{ 1 }-> 1 + A__IF(a__geq(z0, z1), 1 + (1 + (1 + z0 + z1) + (1 + z1)), 0) + A__GEQ(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 A__GEQ(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 A__GEQ(z, z') -{ 1 }-> 1 + A__GEQ(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 A__IF(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 A__IF(z, z', z'') -{ 1 }-> 1 + MARK(z0) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1 A__IF(z, z', z'') -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1 A__MINUS(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 A__MINUS(z, z') -{ 1 }-> 1 + A__MINUS(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 MARK(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__MINUS(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__GEQ(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__IF(mark(z0), z1, z2) + MARK(z0) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__DIV(mark(z0), z1) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__div(z, z') -{ 0 }-> a__if(a__geq(z0, z1), 1 + (1 + (1 + z0 + z1) + (1 + z1)), 0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 a__div(z, z') -{ 0 }-> 0 :|: z0 >= 0, z' = 1 + z0, z = 0 a__div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a__div(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 a__geq(z, z') -{ 0 }-> a__geq(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 a__geq(z, z') -{ 0 }-> 2 :|: z = z0, z0 >= 0, z' = 0 a__geq(z, z') -{ 0 }-> 1 :|: z0 >= 0, z' = 1 + z0, z = 0 a__geq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a__geq(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 a__if(z, z', z'') -{ 0 }-> mark(z0) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z'' = z1 a__if(z, z', z'') -{ 0 }-> mark(z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1 a__if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 a__if(z, z', z'') -{ 0 }-> 1 + z0 + z1 + z2 :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0 a__minus(z, z') -{ 0 }-> a__minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 a__minus(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0 a__minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a__minus(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 mark(z) -{ 0 }-> a__minus(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> a__if(mark(z0), z1, z2) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__geq(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> a__div(mark(z0), z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> 2 :|: z = 2 mark(z) -{ 0 }-> 1 :|: z = 1 mark(z) -{ 0 }-> 0 :|: z = 0 mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 mark(z) -{ 0 }-> 1 + mark(z0) :|: z = 1 + z0, z0 >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) We considered the (Usable) Rules:none And the Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(A__DIV(x_1, x_2)) = 0 POL(A__GEQ(x_1, x_2)) = 0 POL(A__IF(x_1, x_2, x_3)) = x_2 + x_3 POL(A__MINUS(x_1, x_2)) = 0 POL(MARK(x_1)) = x_1 POL(a__div(x_1, x_2)) = [1] + x_1 + x_2 + x_2^2 + x_1*x_2 + x_1^2 POL(a__geq(x_1, x_2)) = 0 POL(a__if(x_1, x_2, x_3)) = [1] + x_2 + x_3 + x_3^2 + x_2*x_3 + [2]x_1^2 + x_2^2 POL(a__minus(x_1, x_2)) = [1] + x_1 + x_2 + x_2^2 + x_1*x_2 + x_1^2 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(div(x_1, x_2)) = x_1 POL(false) = 0 POL(geq(x_1, x_2)) = 0 POL(if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(mark(x_1)) = 0 POL(minus(x_1, x_2)) = 0 POL(s(x_1)) = x_1 POL(true) = 0 ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) K tuples: MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__DIV_2, A__IF_3, MARK_1 Compound Symbols: c1_1, c5_1, c8_2, c10_1, c11_1, c13_1, c14_1, c15_2, c16_2, c18_1 ---------------------------------------- (57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) We considered the (Usable) Rules:none And the Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(A__DIV(x_1, x_2)) = 0 POL(A__GEQ(x_1, x_2)) = 0 POL(A__IF(x_1, x_2, x_3)) = x_2 + x_3 POL(A__MINUS(x_1, x_2)) = 0 POL(MARK(x_1)) = x_1 POL(a__div(x_1, x_2)) = [1] + x_2 POL(a__geq(x_1, x_2)) = 0 POL(a__if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(a__minus(x_1, x_2)) = [1] + x_1 + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(div(x_1, x_2)) = x_1 POL(false) = 0 POL(geq(x_1, x_2)) = [1] + x_1 + x_2 POL(if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(mark(x_1)) = [1] + x_1 POL(minus(x_1, x_2)) = 0 POL(s(x_1)) = x_1 POL(true) = [1] ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) K tuples: MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__DIV_2, A__IF_3, MARK_1 Compound Symbols: c1_1, c5_1, c8_2, c10_1, c11_1, c13_1, c14_1, c15_2, c16_2, c18_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__DIV(s(z0), s(z1)) -> c8(A__IF(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) by A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0), A__GEQ(z0, 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0), A__GEQ(0, s(z0))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(z1)) -> c8(A__IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0), A__GEQ(z0, 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0), A__GEQ(0, s(z0))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(z1)) -> c8(A__IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0), A__GEQ(z0, 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0), A__GEQ(0, s(z0))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(z1)) -> c8(A__IF(geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0), A__GEQ(z0, z1)) K tuples: MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c15_2, c16_2, c18_1, c8_2 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) K tuples: MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c15_2, c16_2, c18_1, c8_2, c8_1 ---------------------------------------- (63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) We considered the (Usable) Rules:none And the Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(A__DIV(x_1, x_2)) = [1] POL(A__GEQ(x_1, x_2)) = 0 POL(A__IF(x_1, x_2, x_3)) = x_2 + x_3 POL(A__MINUS(x_1, x_2)) = 0 POL(MARK(x_1)) = x_1 POL(a__div(x_1, x_2)) = [1] + x_2 POL(a__geq(x_1, x_2)) = 0 POL(a__if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(a__minus(x_1, x_2)) = [1] + x_1 + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(div(x_1, x_2)) = [1] + x_1 POL(false) = [1] POL(geq(x_1, x_2)) = [1] + x_1 + x_2 POL(if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(mark(x_1)) = [1] + x_1 POL(minus(x_1, x_2)) = 0 POL(s(x_1)) = x_1 POL(true) = [1] ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) K tuples: MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c15_2, c16_2, c18_1, c8_2, c8_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(div(z0, z1)) -> c15(A__DIV(mark(z0), z1), MARK(z0)) by MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(0, x1)) -> c15(A__DIV(0, x1), MARK(0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(div(true, x1)) -> c15(A__DIV(true, x1), MARK(true)) MARK(div(false, x1)) -> c15(A__DIV(false, x1), MARK(false)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(0, x1)) -> c15(A__DIV(0, x1), MARK(0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(div(true, x1)) -> c15(A__DIV(true, x1), MARK(true)) MARK(div(false, x1)) -> c15(A__DIV(false, x1), MARK(false)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(0, x1)) -> c15(A__DIV(0, x1), MARK(0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(div(true, x1)) -> c15(A__DIV(true, x1), MARK(true)) MARK(div(false, x1)) -> c15(A__DIV(false, x1), MARK(false)) K tuples: MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c16_2, c18_1, c8_2, c8_1, c15_2 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(div(0, x1)) -> c15(A__DIV(0, x1), MARK(0)) MARK(div(false, x1)) -> c15(A__DIV(false, x1), MARK(false)) MARK(div(true, x1)) -> c15(A__DIV(true, x1), MARK(true)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) K tuples: MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c16_2, c18_1, c8_2, c8_1, c15_2 ---------------------------------------- (69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) We considered the (Usable) Rules:none And the Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(A__DIV(x_1, x_2)) = [1] POL(A__GEQ(x_1, x_2)) = 0 POL(A__IF(x_1, x_2, x_3)) = x_3^2 + x_2^2 POL(A__MINUS(x_1, x_2)) = 0 POL(MARK(x_1)) = x_1^2 POL(a__div(x_1, x_2)) = 0 POL(a__geq(x_1, x_2)) = 0 POL(a__if(x_1, x_2, x_3)) = 0 POL(a__minus(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(div(x_1, x_2)) = [1] + x_1 POL(false) = 0 POL(geq(x_1, x_2)) = [2] POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(mark(x_1)) = 0 POL(minus(x_1, x_2)) = 0 POL(s(x_1)) = x_1 POL(true) = 0 ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) K tuples: MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c16_2, c18_1, c8_2, c8_1, c15_2 ---------------------------------------- (71) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) We considered the (Usable) Rules:none And the Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(A__DIV(x_1, x_2)) = [2] POL(A__GEQ(x_1, x_2)) = 0 POL(A__IF(x_1, x_2, x_3)) = [2]x_3^2 + [2]x_2^2 POL(A__MINUS(x_1, x_2)) = 0 POL(MARK(x_1)) = [2]x_1^2 POL(a__div(x_1, x_2)) = 0 POL(a__geq(x_1, x_2)) = 0 POL(a__if(x_1, x_2, x_3)) = 0 POL(a__minus(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(div(x_1, x_2)) = [1] + x_1 POL(false) = 0 POL(geq(x_1, x_2)) = 0 POL(if(x_1, x_2, x_3)) = [2] + x_1 + x_2 + x_3 POL(mark(x_1)) = 0 POL(minus(x_1, x_2)) = 0 POL(s(x_1)) = x_1 POL(true) = 0 ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) K tuples: MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c16_2, c18_1, c8_2, c8_1, c15_2 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) by MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(0, x1, x2)) -> c16(A__IF(0, x1, x2), MARK(0)) MARK(if(s(z0), x1, x2)) -> c16(A__IF(s(mark(z0)), x1, x2), MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2), MARK(true)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2), MARK(false)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(0, x1, x2)) -> c16(A__IF(0, x1, x2), MARK(0)) MARK(if(s(z0), x1, x2)) -> c16(A__IF(s(mark(z0)), x1, x2), MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2), MARK(true)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2), MARK(false)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) K tuples: MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_2, c8_1, c15_2, c16_2 ---------------------------------------- (75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(if(0, x1, x2)) -> c16(A__IF(0, x1, x2), MARK(0)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(A__IF(s(mark(z0)), x1, x2), MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2), MARK(true)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2), MARK(false)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_2, c8_1, c15_2, c16_2 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_2, c8_1, c15_2, c16_2, c16_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) by A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__IF(geq(z0, z1), s(div(minus(s(z0), s(z1)), s(s(z1)))), 0), A__GEQ(s(z0), s(z1))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2 ---------------------------------------- (81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2 ---------------------------------------- (83) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) We considered the (Usable) Rules:none And the Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(A__DIV(x_1, x_2)) = [1] POL(A__GEQ(x_1, x_2)) = 0 POL(A__IF(x_1, x_2, x_3)) = x_2 + x_3 POL(A__MINUS(x_1, x_2)) = 0 POL(MARK(x_1)) = x_1 POL(a__div(x_1, x_2)) = [1] POL(a__geq(x_1, x_2)) = 0 POL(a__if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(a__minus(x_1, x_2)) = [1] + x_1 + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c16(x_1)) = x_1 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(div(x_1, x_2)) = [1] + x_1 POL(false) = 0 POL(geq(x_1, x_2)) = 0 POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(mark(x_1)) = x_1 POL(minus(x_1, x_2)) = 0 POL(s(x_1)) = x_1 POL(true) = 0 ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(div(minus(z0, z1), x1)) -> c15(A__DIV(a__minus(z0, z1), x1), MARK(minus(z0, z1))) by MARK(div(minus(0, z0), x2)) -> c15(A__DIV(0, x2), MARK(minus(0, z0))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(z0, z1), x2)) -> c15(A__DIV(minus(z0, z1), x2), MARK(minus(z0, z1))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(A__DIV(0, x2), MARK(minus(0, z0))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(z0, z1), x2)) -> c15(A__DIV(minus(z0, z1), x2), MARK(minus(z0, z1))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(A__DIV(0, x2), MARK(minus(0, z0))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(z0, z1), x2)) -> c15(A__DIV(minus(z0, z1), x2), MARK(minus(z0, z1))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2 ---------------------------------------- (87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2, c15_1 ---------------------------------------- (89) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) We considered the (Usable) Rules:none And the Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(A__DIV(x_1, x_2)) = [1] POL(A__GEQ(x_1, x_2)) = 0 POL(A__IF(x_1, x_2, x_3)) = x_2 + x_3 POL(A__MINUS(x_1, x_2)) = 0 POL(MARK(x_1)) = x_1 POL(a__div(x_1, x_2)) = [1] POL(a__geq(x_1, x_2)) = 0 POL(a__if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(a__minus(x_1, x_2)) = [1] + x_1 + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c16(x_1)) = x_1 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(div(x_1, x_2)) = [1] + x_1 POL(false) = 0 POL(geq(x_1, x_2)) = [1] POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(mark(x_1)) = x_1 POL(minus(x_1, x_2)) = 0 POL(s(x_1)) = x_1 POL(true) = 0 ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2, c15_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) by MARK(div(geq(z0, 0), x2)) -> c15(A__DIV(true, x2), MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(A__DIV(false, x2), MARK(geq(0, s(z0)))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(A__DIV(geq(z0, z1), x2), MARK(geq(z0, z1))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(z0, 0), x2)) -> c15(A__DIV(true, x2), MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(A__DIV(false, x2), MARK(geq(0, s(z0)))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(A__DIV(geq(z0, z1), x2), MARK(geq(z0, z1))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2, c15_1 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2, c15_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) by MARK(div(div(x0, z1), x2)) -> c15(A__DIV(div(mark(x0), z1), x2), MARK(div(x0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(0, x1), x2)) -> c15(A__DIV(a__div(0, x1), x2), MARK(div(0, x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2), MARK(div(true, x1))) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2), MARK(div(false, x1))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(x0, z1), x2)) -> c15(A__DIV(div(mark(x0), z1), x2), MARK(div(x0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(0, x1), x2)) -> c15(A__DIV(a__div(0, x1), x2), MARK(div(0, x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2), MARK(div(true, x1))) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2), MARK(div(false, x1))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2, c15_1 ---------------------------------------- (97) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(div(0, x1), x2)) -> c15(A__DIV(a__div(0, x1), x2)) MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2)) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2, c15_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) by MARK(div(if(x0, z1, z2), x3)) -> c15(A__DIV(if(mark(x0), z1, z2), x3), MARK(if(x0, z1, z2))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3), MARK(if(0, x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(div(0, x1), x2)) -> c15(A__DIV(a__div(0, x1), x2)) MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2)) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) MARK(div(if(x0, z1, z2), x3)) -> c15(A__DIV(if(mark(x0), z1, z2), x3), MARK(if(x0, z1, z2))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3), MARK(if(0, x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2, c15_1 ---------------------------------------- (101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(div(0, x1), x2)) -> c15(A__DIV(a__div(0, x1), x2)) MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2)) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2, c15_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(minus(z0, z1), x1, x2)) -> c16(A__IF(a__minus(z0, z1), x1, x2), MARK(minus(z0, z1))) by MARK(if(minus(0, z0), x2, x3)) -> c16(A__IF(0, x2, x3), MARK(minus(0, z0))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(A__IF(minus(z0, z1), x2, x3), MARK(minus(z0, z1))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(div(0, x1), x2)) -> c15(A__DIV(a__div(0, x1), x2)) MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2)) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) MARK(if(minus(0, z0), x2, x3)) -> c16(A__IF(0, x2, x3), MARK(minus(0, z0))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(A__IF(minus(z0, z1), x2, x3), MARK(minus(z0, z1))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2, c15_1 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(div(0, x1), x2)) -> c15(A__DIV(a__div(0, x1), x2)) MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2)) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2, c15_1 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(geq(z0, z1), x1, x2)) -> c16(A__IF(a__geq(z0, z1), x1, x2), MARK(geq(z0, z1))) by MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(A__IF(geq(z0, z1), x2, x3), MARK(geq(z0, z1))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(div(0, x1), x2)) -> c15(A__DIV(a__div(0, x1), x2)) MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2)) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(A__IF(geq(z0, z1), x2, x3), MARK(geq(z0, z1))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2, c15_1 ---------------------------------------- (109) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(div(0, x1), x2)) -> c15(A__DIV(a__div(0, x1), x2)) MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2)) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2, c15_1 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(div(z0, z1), x1, x2)) -> c16(A__IF(a__div(mark(z0), z1), x1, x2), MARK(div(z0, z1))) by MARK(if(div(x0, z1), x2, x3)) -> c16(A__IF(div(mark(x0), z1), x2, x3), MARK(div(x0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(0, x1), x2, x3)) -> c16(A__IF(a__div(0, x1), x2, x3), MARK(div(0, x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3), MARK(div(true, x1))) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3), MARK(div(false, x1))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(div(0, x1), x2)) -> c15(A__DIV(a__div(0, x1), x2)) MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2)) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(x0, z1), x2, x3)) -> c16(A__IF(div(mark(x0), z1), x2, x3), MARK(div(x0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(0, x1), x2, x3)) -> c16(A__IF(a__div(0, x1), x2, x3), MARK(div(0, x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3), MARK(div(true, x1))) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3), MARK(div(false, x1))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2, c15_1 ---------------------------------------- (113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(div(0, x1), x2)) -> c15(A__DIV(a__div(0, x1), x2)) MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2)) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(div(0, x1), x2, x3)) -> c16(A__IF(a__div(0, x1), x2, x3)) MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3)) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_2, c16_1, c8_2, c15_1 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(if(z0, z1, z2), x1, x2)) -> c16(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) by MARK(if(if(x0, z1, z2), x3, x4)) -> c16(A__IF(if(mark(x0), z1, z2), x3, x4), MARK(if(x0, z1, z2))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4), MARK(if(0, x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(div(0, x1), x2)) -> c15(A__DIV(a__div(0, x1), x2)) MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2)) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(div(0, x1), x2, x3)) -> c16(A__IF(a__div(0, x1), x2, x3)) MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3)) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(A__IF(if(mark(x0), z1, z2), x3, x4), MARK(if(x0, z1, z2))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4), MARK(if(0, x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(div(0, x1), x2)) -> c15(A__DIV(a__div(0, x1), x2)) MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2)) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(div(0, x1), x2, x3)) -> c16(A__IF(a__div(0, x1), x2, x3)) MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3)) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(div(div(0, x1), x2)) -> c15(A__DIV(a__div(0, x1), x2)) by MARK(div(div(0, s(z0)), x1)) -> c15(A__DIV(0, x1)) MARK(div(div(0, z1), x1)) -> c15(A__DIV(div(0, z1), x1)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2)) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(div(0, x1), x2, x3)) -> c16(A__IF(a__div(0, x1), x2, x3)) MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3)) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) MARK(div(div(0, s(z0)), x1)) -> c15(A__DIV(0, x1)) MARK(div(div(0, z1), x1)) -> c15(A__DIV(div(0, z1), x1)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(geq(z0, z1), x1)) -> c15(A__DIV(a__geq(z0, z1), x1), MARK(geq(z0, z1))) MARK(div(div(z0, z1), x1)) -> c15(A__DIV(a__div(mark(z0), z1), x1), MARK(div(z0, z1))) MARK(div(if(z0, z1, z2), x1)) -> c15(A__DIV(a__if(mark(z0), z1, z2), x1), MARK(if(z0, z1, z2))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(div(div(0, z1), x1)) -> c15(A__DIV(div(0, z1), x1)) MARK(div(div(0, s(z0)), x1)) -> c15(A__DIV(0, x1)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2)) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(div(0, x1), x2, x3)) -> c16(A__IF(a__div(0, x1), x2, x3)) MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3)) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(div(div(true, x1), x2)) -> c15(A__DIV(a__div(true, x1), x2)) by MARK(div(div(true, z1), x1)) -> c15(A__DIV(div(true, z1), x1)) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(div(0, x1), x2, x3)) -> c16(A__IF(a__div(0, x1), x2, x3)) MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3)) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) MARK(div(div(true, z1), x1)) -> c15(A__DIV(div(true, z1), x1)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (125) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(div(div(true, z1), x1)) -> c15(A__DIV(div(true, z1), x1)) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(div(0, x1), x2, x3)) -> c16(A__IF(a__div(0, x1), x2, x3)) MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3)) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(div(div(false, x1), x2)) -> c15(A__DIV(a__div(false, x1), x2)) by MARK(div(div(false, z1), x1)) -> c15(A__DIV(div(false, z1), x1)) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(div(0, x1), x2, x3)) -> c16(A__IF(a__div(0, x1), x2, x3)) MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3)) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) MARK(div(div(false, z1), x1)) -> c15(A__DIV(div(false, z1), x1)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(div(div(false, z1), x1)) -> c15(A__DIV(div(false, z1), x1)) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(div(0, x1), x2, x3)) -> c16(A__IF(a__div(0, x1), x2, x3)) MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3)) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(div(if(0, x1, x2), x3)) -> c15(A__DIV(a__if(0, x1, x2), x3)) by MARK(div(if(0, z1, z2), x2)) -> c15(A__DIV(if(0, z1, z2), x2)) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(div(0, x1), x2, x3)) -> c16(A__IF(a__div(0, x1), x2, x3)) MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3)) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) MARK(div(if(0, z1, z2), x2)) -> c15(A__DIV(if(0, z1, z2), x2)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (133) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(div(if(0, z1, z2), x2)) -> c15(A__DIV(if(0, z1, z2), x2)) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(div(0, x1), x2, x3)) -> c16(A__IF(a__div(0, x1), x2, x3)) MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3)) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(div(0, x1), x2, x3)) -> c16(A__IF(a__div(0, x1), x2, x3)) by MARK(if(div(0, s(z0)), x1, x2)) -> c16(A__IF(0, x1, x2)) MARK(if(div(0, z1), x1, x2)) -> c16(A__IF(div(0, z1), x1, x2)) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3)) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) MARK(if(div(0, s(z0)), x1, x2)) -> c16(A__IF(0, x1, x2)) MARK(if(div(0, z1), x1, x2)) -> c16(A__IF(div(0, z1), x1, x2)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (137) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(if(div(0, z1), x1, x2)) -> c16(A__IF(div(0, z1), x1, x2)) MARK(if(div(0, s(z0)), x1, x2)) -> c16(A__IF(0, x1, x2)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3)) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(div(true, x1), x2, x3)) -> c16(A__IF(a__div(true, x1), x2, x3)) by MARK(if(div(true, z1), x1, x2)) -> c16(A__IF(div(true, z1), x1, x2)) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) MARK(if(div(true, z1), x1, x2)) -> c16(A__IF(div(true, z1), x1, x2)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (141) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(if(div(true, z1), x1, x2)) -> c16(A__IF(div(true, z1), x1, x2)) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(div(false, x1), x2, x3)) -> c16(A__IF(a__div(false, x1), x2, x3)) by MARK(if(div(false, z1), x1, x2)) -> c16(A__IF(div(false, z1), x1, x2)) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) MARK(if(div(false, z1), x1, x2)) -> c16(A__IF(div(false, z1), x1, x2)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (145) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(if(div(false, z1), x1, x2)) -> c16(A__IF(div(false, z1), x1, x2)) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (147) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(if(0, x1, x2), x3, x4)) -> c16(A__IF(a__if(0, x1, x2), x3, x4)) by MARK(if(if(0, z1, z2), x2, x3)) -> c16(A__IF(if(0, z1, z2), x2, x3)) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) MARK(if(if(0, z1, z2), x2, x3)) -> c16(A__IF(if(0, z1, z2), x2, x3)) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (149) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(if(if(0, z1, z2), x2, x3)) -> c16(A__IF(if(0, z1, z2), x2, x3)) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) S tuples: A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__MINUS_2, A__GEQ_2, A__IF_3, MARK_1, A__DIV_2 Compound Symbols: c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2 ---------------------------------------- (151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__MINUS(s(z0), s(z1)) -> c1(A__MINUS(z0, z1)) by A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) S tuples: A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__GEQ_2, A__IF_3, MARK_1, A__DIV_2, A__MINUS_2 Compound Symbols: c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__GEQ(s(z0), s(z1)) -> c5(A__GEQ(z0, z1)) by A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) S tuples: A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0), A__GEQ(s(z0), s(0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0), A__GEQ(s(0), s(s(z0)))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__IF_3, MARK_1, A__DIV_2, A__MINUS_2, A__GEQ_2 Compound Symbols: c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1, c5_1 ---------------------------------------- (155) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0)) S tuples: A__IF(true, z0, z1) -> c10(MARK(z0)) A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0)) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__IF_3, MARK_1, A__DIV_2, A__MINUS_2, A__GEQ_2 Compound Symbols: c10_1, c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1, c5_1 ---------------------------------------- (157) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__IF(true, z0, z1) -> c10(MARK(z0)) by A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) S tuples: A__IF(false, z0, z1) -> c11(MARK(z1)) MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__IF_3, MARK_1, A__DIV_2, A__MINUS_2, A__GEQ_2 Compound Symbols: c11_1, c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1, c5_1, c10_1 ---------------------------------------- (159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__IF(false, z0, z1) -> c11(MARK(z1)) by A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) S tuples: MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(0), s(s(z0))) -> c8(A__IF(false, s(div(minus(0, s(z0)), s(s(z0)))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__DIV(s(s(0)), s(s(s(z0)))) -> c8(A__IF(false, s(div(minus(s(0), s(s(z0))), s(s(s(z0))))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: MARK_1, A__DIV_2, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1, c5_1, c10_1, c11_1 ---------------------------------------- (161) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) A__DIV(s(0), s(s(z0))) -> c8 A__DIV(s(s(0)), s(s(s(z0)))) -> c8 S tuples: MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) A__DIV(s(0), s(s(z0))) -> c8 A__DIV(s(s(0)), s(s(s(z0)))) -> c8 K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: MARK_1, A__DIV_2, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1, c5_1, c10_1, c11_1, c8 ---------------------------------------- (163) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. A__DIV(s(0), s(s(z0))) -> c8 A__DIV(s(s(0)), s(s(s(z0)))) -> c8 We considered the (Usable) Rules:none And the Tuples: MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) A__DIV(s(0), s(s(z0))) -> c8 A__DIV(s(s(0)), s(s(s(z0)))) -> c8 The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(A__DIV(x_1, x_2)) = [1] POL(A__GEQ(x_1, x_2)) = 0 POL(A__IF(x_1, x_2, x_3)) = x_2 + x_3 POL(A__MINUS(x_1, x_2)) = 0 POL(MARK(x_1)) = x_1 POL(a__div(x_1, x_2)) = 0 POL(a__geq(x_1, x_2)) = 0 POL(a__if(x_1, x_2, x_3)) = x_2 + x_3 POL(a__minus(x_1, x_2)) = x_1 + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c16(x_1)) = x_1 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8) = 0 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(div(x_1, x_2)) = [1] + x_1 POL(false) = 0 POL(geq(x_1, x_2)) = 0 POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(mark(x_1)) = x_1 POL(minus(x_1, x_2)) = 0 POL(s(x_1)) = x_1 POL(true) = 0 ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) A__DIV(s(0), s(s(z0))) -> c8 A__DIV(s(s(0)), s(s(s(z0)))) -> c8 S tuples: MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) A__DIV(s(0), s(s(z0))) -> c8 A__DIV(s(s(0)), s(s(s(z0)))) -> c8 Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: MARK_1, A__DIV_2, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c13_1, c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1, c5_1, c10_1, c11_1, c8 ---------------------------------------- (165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(minus(z0, z1)) -> c13(A__MINUS(z0, z1)) by MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) A__DIV(s(0), s(s(z0))) -> c8 A__DIV(s(s(0)), s(s(s(z0)))) -> c8 MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) S tuples: MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) A__DIV(s(0), s(s(z0))) -> c8 A__DIV(s(s(0)), s(s(s(z0)))) -> c8 Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: MARK_1, A__DIV_2, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1, c5_1, c10_1, c11_1, c8, c13_1 ---------------------------------------- (167) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) A__DIV(s(s(0)), s(s(s(z0)))) -> c8 A__DIV(s(0), s(s(z0))) -> c8 MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) S tuples: MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) K tuples: MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: MARK_1, A__DIV_2, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c14_1, c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1, c5_1, c10_1, c11_1, c13_1 ---------------------------------------- (169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) by MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) S tuples: MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) K tuples: A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: MARK_1, A__DIV_2, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1 ---------------------------------------- (171) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3), MARK(geq(z0, 0))) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3), MARK(geq(0, s(z0)))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) S tuples: MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) K tuples: A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: MARK_1, A__DIV_2, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1 ---------------------------------------- (173) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) S tuples: MARK(s(z0)) -> c18(MARK(z0)) A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) K tuples: A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: MARK_1, A__DIV_2, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c18_1, c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1 ---------------------------------------- (175) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(s(z0)) -> c18(MARK(z0)) by MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) K tuples: A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1 ---------------------------------------- (177) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__DIV(s(z0), s(z1)) -> c8(A__GEQ(z0, z1)) by A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) K tuples: A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1 ---------------------------------------- (179) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(if(s(z0), x1, x2)) -> c16(MARK(s(z0))) by MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) K tuples: A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1 ---------------------------------------- (181) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(if(true, x1, x2)) -> c16(A__IF(true, x1, x2)) by MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) K tuples: A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c16_1, c8_2, c15_1, c16_2, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1 ---------------------------------------- (183) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(if(false, x1, x2)) -> c16(A__IF(false, x1, x2)) by MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) K tuples: A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c15_1, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1 ---------------------------------------- (185) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__DIV(s(s(z0)), s(s(z1))) -> c8(A__GEQ(s(z0), s(z1))) by A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) K tuples: MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c15_1, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1 ---------------------------------------- (187) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(div(minus(s(z0), s(z1)), x2)) -> c15(A__DIV(a__minus(z0, z1), x2), MARK(minus(s(z0), s(z1)))) by MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c15_1, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1 ---------------------------------------- (189) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(div(minus(z0, z1), x2)) -> c15(MARK(minus(z0, z1))) by MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(0, y0), y1), z1) -> c10(MARK(div(minus(0, y0), y1))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(0, y0), y1)) -> c11(MARK(div(minus(0, y0), y1))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c15_1, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1 ---------------------------------------- (191) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, div(minus(0, y0), y1), z1) -> c10 A__IF(false, z0, div(minus(0, y0), y1)) -> c11 S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, div(minus(0, y0), y1), z1) -> c10 A__IF(false, z0, div(minus(0, y0), y1)) -> c11 K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c15_1, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c10, c11 ---------------------------------------- (193) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. A__IF(true, div(minus(0, y0), y1), z1) -> c10 A__IF(false, z0, div(minus(0, y0), y1)) -> c11 We considered the (Usable) Rules:none And the Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, div(minus(0, y0), y1), z1) -> c10 A__IF(false, z0, div(minus(0, y0), y1)) -> c11 The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(A__DIV(x_1, x_2)) = [1] POL(A__GEQ(x_1, x_2)) = 0 POL(A__IF(x_1, x_2, x_3)) = x_2 + x_3 POL(A__MINUS(x_1, x_2)) = 0 POL(MARK(x_1)) = x_1 POL(a__div(x_1, x_2)) = 0 POL(a__geq(x_1, x_2)) = 0 POL(a__if(x_1, x_2, x_3)) = x_2 + x_3 POL(a__minus(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c10) = 0 POL(c10(x_1)) = x_1 POL(c11) = 0 POL(c11(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c16(x_1)) = x_1 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(div(x_1, x_2)) = [1] + x_1 POL(false) = 0 POL(geq(x_1, x_2)) = 0 POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(mark(x_1)) = x_1 POL(minus(x_1, x_2)) = 0 POL(s(x_1)) = x_1 POL(true) = 0 ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, div(minus(0, y0), y1), z1) -> c10 A__IF(false, z0, div(minus(0, y0), y1)) -> c11 S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, div(minus(0, y0), y1), z1) -> c10 A__IF(false, z0, div(minus(0, y0), y1)) -> c11 Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c15_1, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c10, c11 ---------------------------------------- (195) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(div(geq(s(z0), s(z1)), x2)) -> c15(A__DIV(a__geq(z0, z1), x2), MARK(geq(s(z0), s(z1)))) by MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, div(minus(0, y0), y1), z1) -> c10 A__IF(false, z0, div(minus(0, y0), y1)) -> c11 MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, div(minus(0, y0), y1), z1) -> c10 A__IF(false, z0, div(minus(0, y0), y1)) -> c11 Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c15_1, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c10, c11 ---------------------------------------- (197) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: A__IF(false, z0, div(minus(0, y0), y1)) -> c11 A__IF(true, div(minus(0, y0), y1), z1) -> c10 ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c15_1, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1 ---------------------------------------- (199) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(div(geq(z0, z1), x2)) -> c15(MARK(geq(z0, z1))) by MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10(MARK(div(geq(y0, 0), y1))) A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10(MARK(div(geq(0, s(y0)), y1))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, 0), y1)) -> c11(MARK(div(geq(y0, 0), y1))) A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11(MARK(div(geq(0, s(y0)), y1))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c15_1, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1 ---------------------------------------- (201) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10 A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10 A__IF(false, z0, div(geq(y0, 0), y1)) -> c11 A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11 S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10 A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10 A__IF(false, z0, div(geq(y0, 0), y1)) -> c11 A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11 K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c15_1, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c10, c11 ---------------------------------------- (203) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. A__IF(true, div(geq(y0, 0), y1), z1) -> c10 A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10 A__IF(false, z0, div(geq(y0, 0), y1)) -> c11 A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11 We considered the (Usable) Rules:none And the Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10 A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10 A__IF(false, z0, div(geq(y0, 0), y1)) -> c11 A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11 The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(A__DIV(x_1, x_2)) = 0 POL(A__GEQ(x_1, x_2)) = 0 POL(A__IF(x_1, x_2, x_3)) = x_2 + x_3 POL(A__MINUS(x_1, x_2)) = 0 POL(MARK(x_1)) = x_1 POL(a__div(x_1, x_2)) = 0 POL(a__geq(x_1, x_2)) = 0 POL(a__if(x_1, x_2, x_3)) = x_2 + x_3 POL(a__minus(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c10) = 0 POL(c10(x_1)) = x_1 POL(c11) = 0 POL(c11(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c16(x_1)) = x_1 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(div(x_1, x_2)) = x_1 POL(false) = 0 POL(geq(x_1, x_2)) = [1] POL(if(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(mark(x_1)) = x_1 POL(minus(x_1, x_2)) = 0 POL(s(x_1)) = x_1 POL(true) = 0 ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10 A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10 A__IF(false, z0, div(geq(y0, 0), y1)) -> c11 A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11 S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10 A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10 A__IF(false, z0, div(geq(y0, 0), y1)) -> c11 A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11 Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c15_1, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c10, c11 ---------------------------------------- (205) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(div(div(x0, z1), x2)) -> c15(MARK(div(x0, z1))) by MARK(div(div(s(y0), z1), z2)) -> c15(MARK(div(s(y0), z1))) MARK(div(div(div(minus(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(minus(y0, y1), y2), z1))) MARK(div(div(div(geq(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(geq(y0, y1), y2), z1))) MARK(div(div(div(div(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(div(y0, y1), y2), z1))) MARK(div(div(div(if(y0, y1, y2), y3), z1), z2)) -> c15(MARK(div(div(if(y0, y1, y2), y3), z1))) MARK(div(div(div(s(y0), y1), z1), z2)) -> c15(MARK(div(div(s(y0), y1), z1))) MARK(div(div(div(y0, y1), z1), z2)) -> c15(MARK(div(div(y0, y1), z1))) MARK(div(div(if(minus(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(minus(y0, y1), y2, y3), z1))) MARK(div(div(if(geq(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(geq(y0, y1), y2, y3), z1))) MARK(div(div(if(div(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(div(y0, y1), y2, y3), z1))) MARK(div(div(if(if(y0, y1, y2), y3, y4), z1), z2)) -> c15(MARK(div(if(if(y0, y1, y2), y3, y4), z1))) MARK(div(div(if(s(y0), y1, y2), z1), z2)) -> c15(MARK(div(if(s(y0), y1, y2), z1))) MARK(div(div(if(true, y0, y1), z1), z2)) -> c15(MARK(div(if(true, y0, y1), z1))) MARK(div(div(if(false, y0, y1), z1), z2)) -> c15(MARK(div(if(false, y0, y1), z1))) MARK(div(div(if(y0, y1, y2), z1), z2)) -> c15(MARK(div(if(y0, y1, y2), z1))) MARK(div(div(minus(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(minus(s(s(y0)), s(s(y1))), z1))) MARK(div(div(geq(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(geq(s(s(y0)), s(s(y1))), z1))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10 A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10 A__IF(false, z0, div(geq(y0, 0), y1)) -> c11 A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11 MARK(div(div(s(y0), z1), z2)) -> c15(MARK(div(s(y0), z1))) MARK(div(div(div(minus(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(minus(y0, y1), y2), z1))) MARK(div(div(div(geq(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(geq(y0, y1), y2), z1))) MARK(div(div(div(div(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(div(y0, y1), y2), z1))) MARK(div(div(div(if(y0, y1, y2), y3), z1), z2)) -> c15(MARK(div(div(if(y0, y1, y2), y3), z1))) MARK(div(div(div(s(y0), y1), z1), z2)) -> c15(MARK(div(div(s(y0), y1), z1))) MARK(div(div(div(y0, y1), z1), z2)) -> c15(MARK(div(div(y0, y1), z1))) MARK(div(div(if(minus(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(minus(y0, y1), y2, y3), z1))) MARK(div(div(if(geq(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(geq(y0, y1), y2, y3), z1))) MARK(div(div(if(div(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(div(y0, y1), y2, y3), z1))) MARK(div(div(if(if(y0, y1, y2), y3, y4), z1), z2)) -> c15(MARK(div(if(if(y0, y1, y2), y3, y4), z1))) MARK(div(div(if(s(y0), y1, y2), z1), z2)) -> c15(MARK(div(if(s(y0), y1, y2), z1))) MARK(div(div(if(true, y0, y1), z1), z2)) -> c15(MARK(div(if(true, y0, y1), z1))) MARK(div(div(if(false, y0, y1), z1), z2)) -> c15(MARK(div(if(false, y0, y1), z1))) MARK(div(div(if(y0, y1, y2), z1), z2)) -> c15(MARK(div(if(y0, y1, y2), z1))) MARK(div(div(minus(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(minus(s(s(y0)), s(s(y1))), z1))) MARK(div(div(geq(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(geq(s(s(y0)), s(s(y1))), z1))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, div(geq(y0, 0), y1), z1) -> c10 A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10 A__IF(false, z0, div(geq(y0, 0), y1)) -> c11 A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11 Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c15_1, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c10, c11 ---------------------------------------- (207) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: A__IF(false, z0, div(geq(y0, 0), y1)) -> c11 A__IF(false, z0, div(geq(0, s(y0)), y1)) -> c11 A__IF(true, div(geq(0, s(y0)), y1), z1) -> c10 A__IF(true, div(geq(y0, 0), y1), z1) -> c10 ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(div(s(y0), z1), z2)) -> c15(MARK(div(s(y0), z1))) MARK(div(div(div(minus(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(minus(y0, y1), y2), z1))) MARK(div(div(div(geq(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(geq(y0, y1), y2), z1))) MARK(div(div(div(div(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(div(y0, y1), y2), z1))) MARK(div(div(div(if(y0, y1, y2), y3), z1), z2)) -> c15(MARK(div(div(if(y0, y1, y2), y3), z1))) MARK(div(div(div(s(y0), y1), z1), z2)) -> c15(MARK(div(div(s(y0), y1), z1))) MARK(div(div(div(y0, y1), z1), z2)) -> c15(MARK(div(div(y0, y1), z1))) MARK(div(div(if(minus(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(minus(y0, y1), y2, y3), z1))) MARK(div(div(if(geq(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(geq(y0, y1), y2, y3), z1))) MARK(div(div(if(div(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(div(y0, y1), y2, y3), z1))) MARK(div(div(if(if(y0, y1, y2), y3, y4), z1), z2)) -> c15(MARK(div(if(if(y0, y1, y2), y3, y4), z1))) MARK(div(div(if(s(y0), y1, y2), z1), z2)) -> c15(MARK(div(if(s(y0), y1, y2), z1))) MARK(div(div(if(true, y0, y1), z1), z2)) -> c15(MARK(div(if(true, y0, y1), z1))) MARK(div(div(if(false, y0, y1), z1), z2)) -> c15(MARK(div(if(false, y0, y1), z1))) MARK(div(div(if(y0, y1, y2), z1), z2)) -> c15(MARK(div(if(y0, y1, y2), z1))) MARK(div(div(minus(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(minus(s(s(y0)), s(s(y1))), z1))) MARK(div(div(geq(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(geq(s(s(y0)), s(s(y1))), z1))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c15_1, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1 ---------------------------------------- (209) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(div(if(x0, z1, z2), x3)) -> c15(MARK(if(x0, z1, z2))) by MARK(div(if(minus(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(minus(s(y0), s(y1)), z1, z2))) MARK(div(if(minus(y0, y1), z1, z2), z3)) -> c15(MARK(if(minus(y0, y1), z1, z2))) MARK(div(if(geq(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(geq(s(y0), s(y1)), z1, z2))) MARK(div(if(geq(y0, y1), z1, z2), z3)) -> c15(MARK(if(geq(y0, y1), z1, z2))) MARK(div(if(div(minus(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(minus(y0, y1), y2), z1, z2))) MARK(div(if(div(geq(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(geq(y0, y1), y2), z1, z2))) MARK(div(if(div(div(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(div(y0, y1), y2), z1, z2))) MARK(div(if(div(if(y0, y1, y2), y3), z1, z2), z3)) -> c15(MARK(if(div(if(y0, y1, y2), y3), z1, z2))) MARK(div(if(div(s(y0), y1), z1, z2), z3)) -> c15(MARK(if(div(s(y0), y1), z1, z2))) MARK(div(if(div(y0, y1), z1, z2), z3)) -> c15(MARK(if(div(y0, y1), z1, z2))) MARK(div(if(if(minus(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(minus(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(geq(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(geq(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(div(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(div(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(if(y0, y1, y2), y3, y4), z1, z2), z3)) -> c15(MARK(if(if(if(y0, y1, y2), y3, y4), z1, z2))) MARK(div(if(if(s(y0), y1, y2), z1, z2), z3)) -> c15(MARK(if(if(s(y0), y1, y2), z1, z2))) MARK(div(if(if(true, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(true, y0, y1), z1, z2))) MARK(div(if(if(false, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(false, y0, y1), z1, z2))) MARK(div(if(if(y0, y1, y2), z1, z2), z3)) -> c15(MARK(if(if(y0, y1, y2), z1, z2))) MARK(div(if(geq(y0, 0), z1, z2), z3)) -> c15(MARK(if(geq(y0, 0), z1, z2))) MARK(div(if(geq(0, s(y0)), z1, z2), z3)) -> c15(MARK(if(geq(0, s(y0)), z1, z2))) MARK(div(if(s(s(y0)), z1, z2), z3)) -> c15(MARK(if(s(s(y0)), z1, z2))) MARK(div(if(s(div(s(y0), y1)), z1, z2), z3)) -> c15(MARK(if(s(div(s(y0), y1)), z1, z2))) MARK(div(if(s(if(s(y0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(s(y0), y1, y2)), z1, z2))) MARK(div(if(s(if(true, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(true, y0, y1)), z1, z2))) MARK(div(if(s(if(false, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(false, y0, y1)), z1, z2))) MARK(div(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(minus(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(geq(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(div(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2))) MARK(div(if(s(div(div(s(y0), y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(s(y0), y1), y2)), z1, z2))) MARK(div(if(s(div(div(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2), z3)) -> c15(MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2))) MARK(div(if(s(div(if(s(y0), y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2))) MARK(div(if(s(div(if(true, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(true, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(false, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(false, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(y0, y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2))) MARK(div(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(minus(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2))) MARK(div(if(s(if(div(s(y0), y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2), z3)) -> c15(MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2))) MARK(div(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(if(if(true, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(false, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(if(geq(y0, 0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2))) MARK(div(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2))) MARK(div(if(true, minus(y0, y1), z2), z3)) -> c15(MARK(if(true, minus(y0, y1), z2))) MARK(div(if(true, geq(y0, y1), z2), z3)) -> c15(MARK(if(true, geq(y0, y1), z2))) MARK(div(if(true, s(y0), z2), z3)) -> c15(MARK(if(true, s(y0), z2))) MARK(div(if(true, div(s(y0), y1), z2), z3)) -> c15(MARK(if(true, div(s(y0), y1), z2))) MARK(div(if(true, if(s(y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(s(y0), y1, y2), z2))) MARK(div(if(true, if(true, y0, y1), z2), z3)) -> c15(MARK(if(true, if(true, y0, y1), z2))) MARK(div(if(true, if(false, y0, y1), z2), z3)) -> c15(MARK(if(true, if(false, y0, y1), z2))) MARK(div(if(true, div(minus(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(minus(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(minus(0, y0), y1), z2), z3)) -> c15(MARK(if(true, div(minus(0, y0), y1), z2))) MARK(div(if(true, div(minus(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(minus(y0, y1), y2), z2))) MARK(div(if(true, div(geq(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(geq(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(geq(y0, 0), y1), z2), z3)) -> c15(MARK(if(true, div(geq(y0, 0), y1), z2))) MARK(div(if(true, div(geq(0, s(y0)), y1), z2), z3)) -> c15(MARK(if(true, div(geq(0, s(y0)), y1), z2))) MARK(div(if(true, div(geq(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(geq(y0, y1), y2), z2))) MARK(div(if(true, div(div(minus(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(minus(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(geq(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(geq(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(div(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(div(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(if(y0, y1, y2), y3), y4), z2), z3)) -> c15(MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z2))) MARK(div(if(true, div(div(s(y0), y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(s(y0), y1), y2), z2))) MARK(div(if(true, div(div(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(y0, y1), y2), z2))) MARK(div(if(true, div(if(minus(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(geq(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(div(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2), z3)) -> c15(MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2))) MARK(div(if(true, div(if(s(y0), y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(s(y0), y1, y2), y3), z2))) MARK(div(if(true, div(if(true, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(true, y0, y1), y2), z2))) MARK(div(if(true, div(if(false, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(false, y0, y1), y2), z2))) MARK(div(if(true, div(if(y0, y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(y0, y1, y2), y3), z2))) MARK(div(if(true, if(minus(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(minus(0, y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(minus(0, y0), y1, y2), z2))) MARK(div(if(true, if(minus(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(y0, y1), y2, y3), z2))) MARK(div(if(true, if(geq(y0, 0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(y0, 0), y1, y2), z2))) MARK(div(if(true, if(geq(0, s(y0)), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(0, s(y0)), y1, y2), z2))) MARK(div(if(true, if(geq(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(geq(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(y0, y1), y2, y3), z2))) MARK(div(if(true, if(div(minus(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(geq(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(div(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2))) MARK(div(if(true, if(div(s(y0), y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(s(y0), y1), y2, y3), z2))) MARK(div(if(true, if(div(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2), z3)) -> c15(MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2))) MARK(div(if(true, if(if(s(y0), y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z2))) MARK(div(if(true, if(if(true, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(true, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(false, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(false, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(y0, y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(y0, y1, y2), y3, y4), z2))) MARK(div(if(false, z1, minus(y1, y2)), z3)) -> c15(MARK(if(false, z1, minus(y1, y2)))) MARK(div(if(false, z1, geq(y1, y2)), z3)) -> c15(MARK(if(false, z1, geq(y1, y2)))) MARK(div(if(false, z1, s(y1)), z3)) -> c15(MARK(if(false, z1, s(y1)))) MARK(div(if(false, z1, div(s(y1), y2)), z3)) -> c15(MARK(if(false, z1, div(s(y1), y2)))) MARK(div(if(false, z1, if(s(y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(s(y1), y2, y3)))) MARK(div(if(false, z1, if(true, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(true, y1, y2)))) MARK(div(if(false, z1, if(false, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(false, y1, y2)))) MARK(div(if(false, z1, div(minus(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(minus(0, y1), y2)), z3)) -> c15(MARK(if(false, z1, div(minus(0, y1), y2)))) MARK(div(if(false, z1, div(minus(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(y1, y2), y3)))) MARK(div(if(false, z1, div(geq(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(geq(y1, 0), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, 0), y2)))) MARK(div(if(false, z1, div(geq(0, s(y1)), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(0, s(y1)), y2)))) MARK(div(if(false, z1, div(geq(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, y2), y3)))) MARK(div(if(false, z1, div(div(minus(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(minus(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(geq(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(geq(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(div(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(div(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(if(y1, y2, y3), y4), y5)), z3)) -> c15(MARK(if(false, z1, div(div(if(y1, y2, y3), y4), y5)))) MARK(div(if(false, z1, div(div(s(y1), y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(s(y1), y2), y3)))) MARK(div(if(false, z1, div(div(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(y1, y2), y3)))) MARK(div(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(div(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(div(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)), z3)) -> c15(MARK(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)))) MARK(div(if(false, z1, div(if(s(y1), y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(s(y1), y2, y3), y4)))) MARK(div(if(false, z1, div(if(true, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(true, y1, y2), y3)))) MARK(div(if(false, z1, div(if(false, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(false, y1, y2), y3)))) MARK(div(if(false, z1, div(if(y1, y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(y1, y2, y3), y4)))) MARK(div(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(minus(0, y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(minus(0, y1), y2, y3)))) MARK(div(if(false, z1, if(minus(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, 0), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, 0), y2, y3)))) MARK(div(if(false, z1, if(geq(0, s(y1)), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(0, s(y1)), y2, y3)))) MARK(div(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(div(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(div(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)))) MARK(div(if(false, z1, if(div(s(y1), y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(s(y1), y2), y3, y4)))) MARK(div(if(false, z1, if(div(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)), z3)) -> c15(MARK(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)))) MARK(div(if(false, z1, if(if(s(y1), y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(s(y1), y2, y3), y4, y5)))) MARK(div(if(false, z1, if(if(true, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(true, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(false, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(false, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(y1, y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(y1, y2, y3), y4, y5)))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(div(s(y0), z1), z2)) -> c15(MARK(div(s(y0), z1))) MARK(div(div(div(minus(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(minus(y0, y1), y2), z1))) MARK(div(div(div(geq(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(geq(y0, y1), y2), z1))) MARK(div(div(div(div(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(div(y0, y1), y2), z1))) MARK(div(div(div(if(y0, y1, y2), y3), z1), z2)) -> c15(MARK(div(div(if(y0, y1, y2), y3), z1))) MARK(div(div(div(s(y0), y1), z1), z2)) -> c15(MARK(div(div(s(y0), y1), z1))) MARK(div(div(div(y0, y1), z1), z2)) -> c15(MARK(div(div(y0, y1), z1))) MARK(div(div(if(minus(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(minus(y0, y1), y2, y3), z1))) MARK(div(div(if(geq(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(geq(y0, y1), y2, y3), z1))) MARK(div(div(if(div(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(div(y0, y1), y2, y3), z1))) MARK(div(div(if(if(y0, y1, y2), y3, y4), z1), z2)) -> c15(MARK(div(if(if(y0, y1, y2), y3, y4), z1))) MARK(div(div(if(s(y0), y1, y2), z1), z2)) -> c15(MARK(div(if(s(y0), y1, y2), z1))) MARK(div(div(if(true, y0, y1), z1), z2)) -> c15(MARK(div(if(true, y0, y1), z1))) MARK(div(div(if(false, y0, y1), z1), z2)) -> c15(MARK(div(if(false, y0, y1), z1))) MARK(div(div(if(y0, y1, y2), z1), z2)) -> c15(MARK(div(if(y0, y1, y2), z1))) MARK(div(div(minus(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(minus(s(s(y0)), s(s(y1))), z1))) MARK(div(div(geq(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(geq(s(s(y0)), s(s(y1))), z1))) MARK(div(if(minus(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(minus(s(y0), s(y1)), z1, z2))) MARK(div(if(minus(y0, y1), z1, z2), z3)) -> c15(MARK(if(minus(y0, y1), z1, z2))) MARK(div(if(geq(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(geq(s(y0), s(y1)), z1, z2))) MARK(div(if(geq(y0, y1), z1, z2), z3)) -> c15(MARK(if(geq(y0, y1), z1, z2))) MARK(div(if(div(minus(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(minus(y0, y1), y2), z1, z2))) MARK(div(if(div(geq(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(geq(y0, y1), y2), z1, z2))) MARK(div(if(div(div(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(div(y0, y1), y2), z1, z2))) MARK(div(if(div(if(y0, y1, y2), y3), z1, z2), z3)) -> c15(MARK(if(div(if(y0, y1, y2), y3), z1, z2))) MARK(div(if(div(s(y0), y1), z1, z2), z3)) -> c15(MARK(if(div(s(y0), y1), z1, z2))) MARK(div(if(div(y0, y1), z1, z2), z3)) -> c15(MARK(if(div(y0, y1), z1, z2))) MARK(div(if(if(minus(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(minus(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(geq(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(geq(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(div(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(div(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(if(y0, y1, y2), y3, y4), z1, z2), z3)) -> c15(MARK(if(if(if(y0, y1, y2), y3, y4), z1, z2))) MARK(div(if(if(s(y0), y1, y2), z1, z2), z3)) -> c15(MARK(if(if(s(y0), y1, y2), z1, z2))) MARK(div(if(if(true, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(true, y0, y1), z1, z2))) MARK(div(if(if(false, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(false, y0, y1), z1, z2))) MARK(div(if(if(y0, y1, y2), z1, z2), z3)) -> c15(MARK(if(if(y0, y1, y2), z1, z2))) MARK(div(if(geq(y0, 0), z1, z2), z3)) -> c15(MARK(if(geq(y0, 0), z1, z2))) MARK(div(if(geq(0, s(y0)), z1, z2), z3)) -> c15(MARK(if(geq(0, s(y0)), z1, z2))) MARK(div(if(s(s(y0)), z1, z2), z3)) -> c15(MARK(if(s(s(y0)), z1, z2))) MARK(div(if(s(div(s(y0), y1)), z1, z2), z3)) -> c15(MARK(if(s(div(s(y0), y1)), z1, z2))) MARK(div(if(s(if(s(y0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(s(y0), y1, y2)), z1, z2))) MARK(div(if(s(if(true, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(true, y0, y1)), z1, z2))) MARK(div(if(s(if(false, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(false, y0, y1)), z1, z2))) MARK(div(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(minus(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(geq(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(div(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2))) MARK(div(if(s(div(div(s(y0), y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(s(y0), y1), y2)), z1, z2))) MARK(div(if(s(div(div(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2), z3)) -> c15(MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2))) MARK(div(if(s(div(if(s(y0), y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2))) MARK(div(if(s(div(if(true, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(true, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(false, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(false, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(y0, y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2))) MARK(div(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(minus(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2))) MARK(div(if(s(if(div(s(y0), y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2), z3)) -> c15(MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2))) MARK(div(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(if(if(true, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(false, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(if(geq(y0, 0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2))) MARK(div(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2))) MARK(div(if(true, minus(y0, y1), z2), z3)) -> c15(MARK(if(true, minus(y0, y1), z2))) MARK(div(if(true, geq(y0, y1), z2), z3)) -> c15(MARK(if(true, geq(y0, y1), z2))) MARK(div(if(true, s(y0), z2), z3)) -> c15(MARK(if(true, s(y0), z2))) MARK(div(if(true, div(s(y0), y1), z2), z3)) -> c15(MARK(if(true, div(s(y0), y1), z2))) MARK(div(if(true, if(s(y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(s(y0), y1, y2), z2))) MARK(div(if(true, if(true, y0, y1), z2), z3)) -> c15(MARK(if(true, if(true, y0, y1), z2))) MARK(div(if(true, if(false, y0, y1), z2), z3)) -> c15(MARK(if(true, if(false, y0, y1), z2))) MARK(div(if(true, div(minus(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(minus(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(minus(0, y0), y1), z2), z3)) -> c15(MARK(if(true, div(minus(0, y0), y1), z2))) MARK(div(if(true, div(minus(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(minus(y0, y1), y2), z2))) MARK(div(if(true, div(geq(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(geq(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(geq(y0, 0), y1), z2), z3)) -> c15(MARK(if(true, div(geq(y0, 0), y1), z2))) MARK(div(if(true, div(geq(0, s(y0)), y1), z2), z3)) -> c15(MARK(if(true, div(geq(0, s(y0)), y1), z2))) MARK(div(if(true, div(geq(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(geq(y0, y1), y2), z2))) MARK(div(if(true, div(div(minus(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(minus(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(geq(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(geq(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(div(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(div(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(if(y0, y1, y2), y3), y4), z2), z3)) -> c15(MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z2))) MARK(div(if(true, div(div(s(y0), y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(s(y0), y1), y2), z2))) MARK(div(if(true, div(div(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(y0, y1), y2), z2))) MARK(div(if(true, div(if(minus(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(geq(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(div(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2), z3)) -> c15(MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2))) MARK(div(if(true, div(if(s(y0), y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(s(y0), y1, y2), y3), z2))) MARK(div(if(true, div(if(true, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(true, y0, y1), y2), z2))) MARK(div(if(true, div(if(false, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(false, y0, y1), y2), z2))) MARK(div(if(true, div(if(y0, y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(y0, y1, y2), y3), z2))) MARK(div(if(true, if(minus(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(minus(0, y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(minus(0, y0), y1, y2), z2))) MARK(div(if(true, if(minus(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(y0, y1), y2, y3), z2))) MARK(div(if(true, if(geq(y0, 0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(y0, 0), y1, y2), z2))) MARK(div(if(true, if(geq(0, s(y0)), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(0, s(y0)), y1, y2), z2))) MARK(div(if(true, if(geq(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(geq(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(y0, y1), y2, y3), z2))) MARK(div(if(true, if(div(minus(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(geq(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(div(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2))) MARK(div(if(true, if(div(s(y0), y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(s(y0), y1), y2, y3), z2))) MARK(div(if(true, if(div(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2), z3)) -> c15(MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2))) MARK(div(if(true, if(if(s(y0), y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z2))) MARK(div(if(true, if(if(true, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(true, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(false, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(false, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(y0, y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(y0, y1, y2), y3, y4), z2))) MARK(div(if(false, z1, minus(y1, y2)), z3)) -> c15(MARK(if(false, z1, minus(y1, y2)))) MARK(div(if(false, z1, geq(y1, y2)), z3)) -> c15(MARK(if(false, z1, geq(y1, y2)))) MARK(div(if(false, z1, s(y1)), z3)) -> c15(MARK(if(false, z1, s(y1)))) MARK(div(if(false, z1, div(s(y1), y2)), z3)) -> c15(MARK(if(false, z1, div(s(y1), y2)))) MARK(div(if(false, z1, if(s(y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(s(y1), y2, y3)))) MARK(div(if(false, z1, if(true, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(true, y1, y2)))) MARK(div(if(false, z1, if(false, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(false, y1, y2)))) MARK(div(if(false, z1, div(minus(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(minus(0, y1), y2)), z3)) -> c15(MARK(if(false, z1, div(minus(0, y1), y2)))) MARK(div(if(false, z1, div(minus(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(y1, y2), y3)))) MARK(div(if(false, z1, div(geq(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(geq(y1, 0), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, 0), y2)))) MARK(div(if(false, z1, div(geq(0, s(y1)), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(0, s(y1)), y2)))) MARK(div(if(false, z1, div(geq(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, y2), y3)))) MARK(div(if(false, z1, div(div(minus(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(minus(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(geq(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(geq(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(div(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(div(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(if(y1, y2, y3), y4), y5)), z3)) -> c15(MARK(if(false, z1, div(div(if(y1, y2, y3), y4), y5)))) MARK(div(if(false, z1, div(div(s(y1), y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(s(y1), y2), y3)))) MARK(div(if(false, z1, div(div(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(y1, y2), y3)))) MARK(div(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(div(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(div(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)), z3)) -> c15(MARK(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)))) MARK(div(if(false, z1, div(if(s(y1), y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(s(y1), y2, y3), y4)))) MARK(div(if(false, z1, div(if(true, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(true, y1, y2), y3)))) MARK(div(if(false, z1, div(if(false, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(false, y1, y2), y3)))) MARK(div(if(false, z1, div(if(y1, y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(y1, y2, y3), y4)))) MARK(div(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(minus(0, y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(minus(0, y1), y2, y3)))) MARK(div(if(false, z1, if(minus(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, 0), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, 0), y2, y3)))) MARK(div(if(false, z1, if(geq(0, s(y1)), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(0, s(y1)), y2, y3)))) MARK(div(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(div(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(div(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)))) MARK(div(if(false, z1, if(div(s(y1), y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(s(y1), y2), y3, y4)))) MARK(div(if(false, z1, if(div(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)), z3)) -> c15(MARK(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)))) MARK(div(if(false, z1, if(if(s(y1), y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(s(y1), y2, y3), y4, y5)))) MARK(div(if(false, z1, if(if(true, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(true, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(false, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(false, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(y1, y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(y1, y2, y3), y4, y5)))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c15_1 ---------------------------------------- (211) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(if(minus(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__minus(z0, z1), x2, x3), MARK(minus(s(z0), s(z1)))) by MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(A__IF(a__minus(s(y0), s(y1)), z2, z3), MARK(minus(s(s(y0)), s(s(y1))))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(div(s(y0), z1), z2)) -> c15(MARK(div(s(y0), z1))) MARK(div(div(div(minus(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(minus(y0, y1), y2), z1))) MARK(div(div(div(geq(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(geq(y0, y1), y2), z1))) MARK(div(div(div(div(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(div(y0, y1), y2), z1))) MARK(div(div(div(if(y0, y1, y2), y3), z1), z2)) -> c15(MARK(div(div(if(y0, y1, y2), y3), z1))) MARK(div(div(div(s(y0), y1), z1), z2)) -> c15(MARK(div(div(s(y0), y1), z1))) MARK(div(div(div(y0, y1), z1), z2)) -> c15(MARK(div(div(y0, y1), z1))) MARK(div(div(if(minus(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(minus(y0, y1), y2, y3), z1))) MARK(div(div(if(geq(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(geq(y0, y1), y2, y3), z1))) MARK(div(div(if(div(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(div(y0, y1), y2, y3), z1))) MARK(div(div(if(if(y0, y1, y2), y3, y4), z1), z2)) -> c15(MARK(div(if(if(y0, y1, y2), y3, y4), z1))) MARK(div(div(if(s(y0), y1, y2), z1), z2)) -> c15(MARK(div(if(s(y0), y1, y2), z1))) MARK(div(div(if(true, y0, y1), z1), z2)) -> c15(MARK(div(if(true, y0, y1), z1))) MARK(div(div(if(false, y0, y1), z1), z2)) -> c15(MARK(div(if(false, y0, y1), z1))) MARK(div(div(if(y0, y1, y2), z1), z2)) -> c15(MARK(div(if(y0, y1, y2), z1))) MARK(div(div(minus(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(minus(s(s(y0)), s(s(y1))), z1))) MARK(div(div(geq(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(geq(s(s(y0)), s(s(y1))), z1))) MARK(div(if(minus(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(minus(s(y0), s(y1)), z1, z2))) MARK(div(if(minus(y0, y1), z1, z2), z3)) -> c15(MARK(if(minus(y0, y1), z1, z2))) MARK(div(if(geq(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(geq(s(y0), s(y1)), z1, z2))) MARK(div(if(geq(y0, y1), z1, z2), z3)) -> c15(MARK(if(geq(y0, y1), z1, z2))) MARK(div(if(div(minus(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(minus(y0, y1), y2), z1, z2))) MARK(div(if(div(geq(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(geq(y0, y1), y2), z1, z2))) MARK(div(if(div(div(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(div(y0, y1), y2), z1, z2))) MARK(div(if(div(if(y0, y1, y2), y3), z1, z2), z3)) -> c15(MARK(if(div(if(y0, y1, y2), y3), z1, z2))) MARK(div(if(div(s(y0), y1), z1, z2), z3)) -> c15(MARK(if(div(s(y0), y1), z1, z2))) MARK(div(if(div(y0, y1), z1, z2), z3)) -> c15(MARK(if(div(y0, y1), z1, z2))) MARK(div(if(if(minus(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(minus(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(geq(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(geq(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(div(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(div(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(if(y0, y1, y2), y3, y4), z1, z2), z3)) -> c15(MARK(if(if(if(y0, y1, y2), y3, y4), z1, z2))) MARK(div(if(if(s(y0), y1, y2), z1, z2), z3)) -> c15(MARK(if(if(s(y0), y1, y2), z1, z2))) MARK(div(if(if(true, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(true, y0, y1), z1, z2))) MARK(div(if(if(false, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(false, y0, y1), z1, z2))) MARK(div(if(if(y0, y1, y2), z1, z2), z3)) -> c15(MARK(if(if(y0, y1, y2), z1, z2))) MARK(div(if(geq(y0, 0), z1, z2), z3)) -> c15(MARK(if(geq(y0, 0), z1, z2))) MARK(div(if(geq(0, s(y0)), z1, z2), z3)) -> c15(MARK(if(geq(0, s(y0)), z1, z2))) MARK(div(if(s(s(y0)), z1, z2), z3)) -> c15(MARK(if(s(s(y0)), z1, z2))) MARK(div(if(s(div(s(y0), y1)), z1, z2), z3)) -> c15(MARK(if(s(div(s(y0), y1)), z1, z2))) MARK(div(if(s(if(s(y0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(s(y0), y1, y2)), z1, z2))) MARK(div(if(s(if(true, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(true, y0, y1)), z1, z2))) MARK(div(if(s(if(false, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(false, y0, y1)), z1, z2))) MARK(div(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(minus(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(geq(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(div(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2))) MARK(div(if(s(div(div(s(y0), y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(s(y0), y1), y2)), z1, z2))) MARK(div(if(s(div(div(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2), z3)) -> c15(MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2))) MARK(div(if(s(div(if(s(y0), y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2))) MARK(div(if(s(div(if(true, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(true, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(false, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(false, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(y0, y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2))) MARK(div(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(minus(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2))) MARK(div(if(s(if(div(s(y0), y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2), z3)) -> c15(MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2))) MARK(div(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(if(if(true, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(false, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(if(geq(y0, 0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2))) MARK(div(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2))) MARK(div(if(true, minus(y0, y1), z2), z3)) -> c15(MARK(if(true, minus(y0, y1), z2))) MARK(div(if(true, geq(y0, y1), z2), z3)) -> c15(MARK(if(true, geq(y0, y1), z2))) MARK(div(if(true, s(y0), z2), z3)) -> c15(MARK(if(true, s(y0), z2))) MARK(div(if(true, div(s(y0), y1), z2), z3)) -> c15(MARK(if(true, div(s(y0), y1), z2))) MARK(div(if(true, if(s(y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(s(y0), y1, y2), z2))) MARK(div(if(true, if(true, y0, y1), z2), z3)) -> c15(MARK(if(true, if(true, y0, y1), z2))) MARK(div(if(true, if(false, y0, y1), z2), z3)) -> c15(MARK(if(true, if(false, y0, y1), z2))) MARK(div(if(true, div(minus(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(minus(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(minus(0, y0), y1), z2), z3)) -> c15(MARK(if(true, div(minus(0, y0), y1), z2))) MARK(div(if(true, div(minus(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(minus(y0, y1), y2), z2))) MARK(div(if(true, div(geq(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(geq(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(geq(y0, 0), y1), z2), z3)) -> c15(MARK(if(true, div(geq(y0, 0), y1), z2))) MARK(div(if(true, div(geq(0, s(y0)), y1), z2), z3)) -> c15(MARK(if(true, div(geq(0, s(y0)), y1), z2))) MARK(div(if(true, div(geq(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(geq(y0, y1), y2), z2))) MARK(div(if(true, div(div(minus(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(minus(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(geq(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(geq(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(div(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(div(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(if(y0, y1, y2), y3), y4), z2), z3)) -> c15(MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z2))) MARK(div(if(true, div(div(s(y0), y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(s(y0), y1), y2), z2))) MARK(div(if(true, div(div(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(y0, y1), y2), z2))) MARK(div(if(true, div(if(minus(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(geq(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(div(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2), z3)) -> c15(MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2))) MARK(div(if(true, div(if(s(y0), y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(s(y0), y1, y2), y3), z2))) MARK(div(if(true, div(if(true, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(true, y0, y1), y2), z2))) MARK(div(if(true, div(if(false, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(false, y0, y1), y2), z2))) MARK(div(if(true, div(if(y0, y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(y0, y1, y2), y3), z2))) MARK(div(if(true, if(minus(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(minus(0, y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(minus(0, y0), y1, y2), z2))) MARK(div(if(true, if(minus(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(y0, y1), y2, y3), z2))) MARK(div(if(true, if(geq(y0, 0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(y0, 0), y1, y2), z2))) MARK(div(if(true, if(geq(0, s(y0)), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(0, s(y0)), y1, y2), z2))) MARK(div(if(true, if(geq(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(geq(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(y0, y1), y2, y3), z2))) MARK(div(if(true, if(div(minus(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(geq(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(div(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2))) MARK(div(if(true, if(div(s(y0), y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(s(y0), y1), y2, y3), z2))) MARK(div(if(true, if(div(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2), z3)) -> c15(MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2))) MARK(div(if(true, if(if(s(y0), y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z2))) MARK(div(if(true, if(if(true, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(true, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(false, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(false, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(y0, y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(y0, y1, y2), y3, y4), z2))) MARK(div(if(false, z1, minus(y1, y2)), z3)) -> c15(MARK(if(false, z1, minus(y1, y2)))) MARK(div(if(false, z1, geq(y1, y2)), z3)) -> c15(MARK(if(false, z1, geq(y1, y2)))) MARK(div(if(false, z1, s(y1)), z3)) -> c15(MARK(if(false, z1, s(y1)))) MARK(div(if(false, z1, div(s(y1), y2)), z3)) -> c15(MARK(if(false, z1, div(s(y1), y2)))) MARK(div(if(false, z1, if(s(y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(s(y1), y2, y3)))) MARK(div(if(false, z1, if(true, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(true, y1, y2)))) MARK(div(if(false, z1, if(false, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(false, y1, y2)))) MARK(div(if(false, z1, div(minus(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(minus(0, y1), y2)), z3)) -> c15(MARK(if(false, z1, div(minus(0, y1), y2)))) MARK(div(if(false, z1, div(minus(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(y1, y2), y3)))) MARK(div(if(false, z1, div(geq(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(geq(y1, 0), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, 0), y2)))) MARK(div(if(false, z1, div(geq(0, s(y1)), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(0, s(y1)), y2)))) MARK(div(if(false, z1, div(geq(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, y2), y3)))) MARK(div(if(false, z1, div(div(minus(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(minus(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(geq(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(geq(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(div(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(div(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(if(y1, y2, y3), y4), y5)), z3)) -> c15(MARK(if(false, z1, div(div(if(y1, y2, y3), y4), y5)))) MARK(div(if(false, z1, div(div(s(y1), y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(s(y1), y2), y3)))) MARK(div(if(false, z1, div(div(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(y1, y2), y3)))) MARK(div(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(div(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(div(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)), z3)) -> c15(MARK(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)))) MARK(div(if(false, z1, div(if(s(y1), y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(s(y1), y2, y3), y4)))) MARK(div(if(false, z1, div(if(true, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(true, y1, y2), y3)))) MARK(div(if(false, z1, div(if(false, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(false, y1, y2), y3)))) MARK(div(if(false, z1, div(if(y1, y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(y1, y2, y3), y4)))) MARK(div(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(minus(0, y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(minus(0, y1), y2, y3)))) MARK(div(if(false, z1, if(minus(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, 0), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, 0), y2, y3)))) MARK(div(if(false, z1, if(geq(0, s(y1)), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(0, s(y1)), y2, y3)))) MARK(div(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(div(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(div(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)))) MARK(div(if(false, z1, if(div(s(y1), y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(s(y1), y2), y3, y4)))) MARK(div(if(false, z1, if(div(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)), z3)) -> c15(MARK(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)))) MARK(div(if(false, z1, if(if(s(y1), y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(s(y1), y2, y3), y4, y5)))) MARK(div(if(false, z1, if(if(true, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(true, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(false, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(false, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(y1, y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(y1, y2, y3), y4, y5)))) MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(A__IF(a__minus(s(y0), s(y1)), z2, z3), MARK(minus(s(s(y0)), s(s(y1))))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c16_1, c16_2, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c15_1 ---------------------------------------- (213) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(if(minus(z0, z1), x2, x3)) -> c16(MARK(minus(z0, z1))) by MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(MARK(minus(s(s(y0)), s(s(y1))))) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(div(s(y0), z1), z2)) -> c15(MARK(div(s(y0), z1))) MARK(div(div(div(minus(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(minus(y0, y1), y2), z1))) MARK(div(div(div(geq(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(geq(y0, y1), y2), z1))) MARK(div(div(div(div(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(div(y0, y1), y2), z1))) MARK(div(div(div(if(y0, y1, y2), y3), z1), z2)) -> c15(MARK(div(div(if(y0, y1, y2), y3), z1))) MARK(div(div(div(s(y0), y1), z1), z2)) -> c15(MARK(div(div(s(y0), y1), z1))) MARK(div(div(div(y0, y1), z1), z2)) -> c15(MARK(div(div(y0, y1), z1))) MARK(div(div(if(minus(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(minus(y0, y1), y2, y3), z1))) MARK(div(div(if(geq(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(geq(y0, y1), y2, y3), z1))) MARK(div(div(if(div(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(div(y0, y1), y2, y3), z1))) MARK(div(div(if(if(y0, y1, y2), y3, y4), z1), z2)) -> c15(MARK(div(if(if(y0, y1, y2), y3, y4), z1))) MARK(div(div(if(s(y0), y1, y2), z1), z2)) -> c15(MARK(div(if(s(y0), y1, y2), z1))) MARK(div(div(if(true, y0, y1), z1), z2)) -> c15(MARK(div(if(true, y0, y1), z1))) MARK(div(div(if(false, y0, y1), z1), z2)) -> c15(MARK(div(if(false, y0, y1), z1))) MARK(div(div(if(y0, y1, y2), z1), z2)) -> c15(MARK(div(if(y0, y1, y2), z1))) MARK(div(div(minus(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(minus(s(s(y0)), s(s(y1))), z1))) MARK(div(div(geq(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(geq(s(s(y0)), s(s(y1))), z1))) MARK(div(if(minus(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(minus(s(y0), s(y1)), z1, z2))) MARK(div(if(minus(y0, y1), z1, z2), z3)) -> c15(MARK(if(minus(y0, y1), z1, z2))) MARK(div(if(geq(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(geq(s(y0), s(y1)), z1, z2))) MARK(div(if(geq(y0, y1), z1, z2), z3)) -> c15(MARK(if(geq(y0, y1), z1, z2))) MARK(div(if(div(minus(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(minus(y0, y1), y2), z1, z2))) MARK(div(if(div(geq(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(geq(y0, y1), y2), z1, z2))) MARK(div(if(div(div(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(div(y0, y1), y2), z1, z2))) MARK(div(if(div(if(y0, y1, y2), y3), z1, z2), z3)) -> c15(MARK(if(div(if(y0, y1, y2), y3), z1, z2))) MARK(div(if(div(s(y0), y1), z1, z2), z3)) -> c15(MARK(if(div(s(y0), y1), z1, z2))) MARK(div(if(div(y0, y1), z1, z2), z3)) -> c15(MARK(if(div(y0, y1), z1, z2))) MARK(div(if(if(minus(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(minus(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(geq(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(geq(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(div(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(div(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(if(y0, y1, y2), y3, y4), z1, z2), z3)) -> c15(MARK(if(if(if(y0, y1, y2), y3, y4), z1, z2))) MARK(div(if(if(s(y0), y1, y2), z1, z2), z3)) -> c15(MARK(if(if(s(y0), y1, y2), z1, z2))) MARK(div(if(if(true, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(true, y0, y1), z1, z2))) MARK(div(if(if(false, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(false, y0, y1), z1, z2))) MARK(div(if(if(y0, y1, y2), z1, z2), z3)) -> c15(MARK(if(if(y0, y1, y2), z1, z2))) MARK(div(if(geq(y0, 0), z1, z2), z3)) -> c15(MARK(if(geq(y0, 0), z1, z2))) MARK(div(if(geq(0, s(y0)), z1, z2), z3)) -> c15(MARK(if(geq(0, s(y0)), z1, z2))) MARK(div(if(s(s(y0)), z1, z2), z3)) -> c15(MARK(if(s(s(y0)), z1, z2))) MARK(div(if(s(div(s(y0), y1)), z1, z2), z3)) -> c15(MARK(if(s(div(s(y0), y1)), z1, z2))) MARK(div(if(s(if(s(y0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(s(y0), y1, y2)), z1, z2))) MARK(div(if(s(if(true, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(true, y0, y1)), z1, z2))) MARK(div(if(s(if(false, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(false, y0, y1)), z1, z2))) MARK(div(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(minus(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(geq(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(div(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2))) MARK(div(if(s(div(div(s(y0), y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(s(y0), y1), y2)), z1, z2))) MARK(div(if(s(div(div(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2), z3)) -> c15(MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2))) MARK(div(if(s(div(if(s(y0), y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2))) MARK(div(if(s(div(if(true, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(true, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(false, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(false, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(y0, y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2))) MARK(div(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(minus(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2))) MARK(div(if(s(if(div(s(y0), y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2), z3)) -> c15(MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2))) MARK(div(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(if(if(true, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(false, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(if(geq(y0, 0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2))) MARK(div(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2))) MARK(div(if(true, minus(y0, y1), z2), z3)) -> c15(MARK(if(true, minus(y0, y1), z2))) MARK(div(if(true, geq(y0, y1), z2), z3)) -> c15(MARK(if(true, geq(y0, y1), z2))) MARK(div(if(true, s(y0), z2), z3)) -> c15(MARK(if(true, s(y0), z2))) MARK(div(if(true, div(s(y0), y1), z2), z3)) -> c15(MARK(if(true, div(s(y0), y1), z2))) MARK(div(if(true, if(s(y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(s(y0), y1, y2), z2))) MARK(div(if(true, if(true, y0, y1), z2), z3)) -> c15(MARK(if(true, if(true, y0, y1), z2))) MARK(div(if(true, if(false, y0, y1), z2), z3)) -> c15(MARK(if(true, if(false, y0, y1), z2))) MARK(div(if(true, div(minus(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(minus(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(minus(0, y0), y1), z2), z3)) -> c15(MARK(if(true, div(minus(0, y0), y1), z2))) MARK(div(if(true, div(minus(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(minus(y0, y1), y2), z2))) MARK(div(if(true, div(geq(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(geq(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(geq(y0, 0), y1), z2), z3)) -> c15(MARK(if(true, div(geq(y0, 0), y1), z2))) MARK(div(if(true, div(geq(0, s(y0)), y1), z2), z3)) -> c15(MARK(if(true, div(geq(0, s(y0)), y1), z2))) MARK(div(if(true, div(geq(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(geq(y0, y1), y2), z2))) MARK(div(if(true, div(div(minus(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(minus(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(geq(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(geq(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(div(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(div(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(if(y0, y1, y2), y3), y4), z2), z3)) -> c15(MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z2))) MARK(div(if(true, div(div(s(y0), y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(s(y0), y1), y2), z2))) MARK(div(if(true, div(div(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(y0, y1), y2), z2))) MARK(div(if(true, div(if(minus(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(geq(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(div(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2), z3)) -> c15(MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2))) MARK(div(if(true, div(if(s(y0), y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(s(y0), y1, y2), y3), z2))) MARK(div(if(true, div(if(true, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(true, y0, y1), y2), z2))) MARK(div(if(true, div(if(false, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(false, y0, y1), y2), z2))) MARK(div(if(true, div(if(y0, y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(y0, y1, y2), y3), z2))) MARK(div(if(true, if(minus(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(minus(0, y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(minus(0, y0), y1, y2), z2))) MARK(div(if(true, if(minus(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(y0, y1), y2, y3), z2))) MARK(div(if(true, if(geq(y0, 0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(y0, 0), y1, y2), z2))) MARK(div(if(true, if(geq(0, s(y0)), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(0, s(y0)), y1, y2), z2))) MARK(div(if(true, if(geq(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(geq(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(y0, y1), y2, y3), z2))) MARK(div(if(true, if(div(minus(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(geq(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(div(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2))) MARK(div(if(true, if(div(s(y0), y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(s(y0), y1), y2, y3), z2))) MARK(div(if(true, if(div(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2), z3)) -> c15(MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2))) MARK(div(if(true, if(if(s(y0), y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z2))) MARK(div(if(true, if(if(true, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(true, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(false, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(false, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(y0, y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(y0, y1, y2), y3, y4), z2))) MARK(div(if(false, z1, minus(y1, y2)), z3)) -> c15(MARK(if(false, z1, minus(y1, y2)))) MARK(div(if(false, z1, geq(y1, y2)), z3)) -> c15(MARK(if(false, z1, geq(y1, y2)))) MARK(div(if(false, z1, s(y1)), z3)) -> c15(MARK(if(false, z1, s(y1)))) MARK(div(if(false, z1, div(s(y1), y2)), z3)) -> c15(MARK(if(false, z1, div(s(y1), y2)))) MARK(div(if(false, z1, if(s(y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(s(y1), y2, y3)))) MARK(div(if(false, z1, if(true, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(true, y1, y2)))) MARK(div(if(false, z1, if(false, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(false, y1, y2)))) MARK(div(if(false, z1, div(minus(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(minus(0, y1), y2)), z3)) -> c15(MARK(if(false, z1, div(minus(0, y1), y2)))) MARK(div(if(false, z1, div(minus(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(y1, y2), y3)))) MARK(div(if(false, z1, div(geq(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(geq(y1, 0), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, 0), y2)))) MARK(div(if(false, z1, div(geq(0, s(y1)), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(0, s(y1)), y2)))) MARK(div(if(false, z1, div(geq(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, y2), y3)))) MARK(div(if(false, z1, div(div(minus(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(minus(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(geq(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(geq(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(div(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(div(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(if(y1, y2, y3), y4), y5)), z3)) -> c15(MARK(if(false, z1, div(div(if(y1, y2, y3), y4), y5)))) MARK(div(if(false, z1, div(div(s(y1), y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(s(y1), y2), y3)))) MARK(div(if(false, z1, div(div(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(y1, y2), y3)))) MARK(div(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(div(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(div(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)), z3)) -> c15(MARK(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)))) MARK(div(if(false, z1, div(if(s(y1), y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(s(y1), y2, y3), y4)))) MARK(div(if(false, z1, div(if(true, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(true, y1, y2), y3)))) MARK(div(if(false, z1, div(if(false, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(false, y1, y2), y3)))) MARK(div(if(false, z1, div(if(y1, y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(y1, y2, y3), y4)))) MARK(div(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(minus(0, y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(minus(0, y1), y2, y3)))) MARK(div(if(false, z1, if(minus(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, 0), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, 0), y2, y3)))) MARK(div(if(false, z1, if(geq(0, s(y1)), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(0, s(y1)), y2, y3)))) MARK(div(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(div(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(div(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)))) MARK(div(if(false, z1, if(div(s(y1), y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(s(y1), y2), y3, y4)))) MARK(div(if(false, z1, if(div(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)), z3)) -> c15(MARK(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)))) MARK(div(if(false, z1, if(if(s(y1), y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(s(y1), y2, y3), y4, y5)))) MARK(div(if(false, z1, if(if(true, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(true, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(false, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(false, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(y1, y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(y1, y2, y3), y4, y5)))) MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(A__IF(a__minus(s(y0), s(y1)), z2, z3), MARK(minus(s(s(y0)), s(s(y1))))) MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(MARK(minus(s(s(y0)), s(s(y1))))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10(MARK(if(minus(0, y0), y1, y2))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11(MARK(if(minus(0, y0), y1, y2))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c15_1 ---------------------------------------- (215) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(div(s(y0), z1), z2)) -> c15(MARK(div(s(y0), z1))) MARK(div(div(div(minus(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(minus(y0, y1), y2), z1))) MARK(div(div(div(geq(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(geq(y0, y1), y2), z1))) MARK(div(div(div(div(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(div(y0, y1), y2), z1))) MARK(div(div(div(if(y0, y1, y2), y3), z1), z2)) -> c15(MARK(div(div(if(y0, y1, y2), y3), z1))) MARK(div(div(div(s(y0), y1), z1), z2)) -> c15(MARK(div(div(s(y0), y1), z1))) MARK(div(div(div(y0, y1), z1), z2)) -> c15(MARK(div(div(y0, y1), z1))) MARK(div(div(if(minus(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(minus(y0, y1), y2, y3), z1))) MARK(div(div(if(geq(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(geq(y0, y1), y2, y3), z1))) MARK(div(div(if(div(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(div(y0, y1), y2, y3), z1))) MARK(div(div(if(if(y0, y1, y2), y3, y4), z1), z2)) -> c15(MARK(div(if(if(y0, y1, y2), y3, y4), z1))) MARK(div(div(if(s(y0), y1, y2), z1), z2)) -> c15(MARK(div(if(s(y0), y1, y2), z1))) MARK(div(div(if(true, y0, y1), z1), z2)) -> c15(MARK(div(if(true, y0, y1), z1))) MARK(div(div(if(false, y0, y1), z1), z2)) -> c15(MARK(div(if(false, y0, y1), z1))) MARK(div(div(if(y0, y1, y2), z1), z2)) -> c15(MARK(div(if(y0, y1, y2), z1))) MARK(div(div(minus(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(minus(s(s(y0)), s(s(y1))), z1))) MARK(div(div(geq(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(geq(s(s(y0)), s(s(y1))), z1))) MARK(div(if(minus(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(minus(s(y0), s(y1)), z1, z2))) MARK(div(if(minus(y0, y1), z1, z2), z3)) -> c15(MARK(if(minus(y0, y1), z1, z2))) MARK(div(if(geq(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(geq(s(y0), s(y1)), z1, z2))) MARK(div(if(geq(y0, y1), z1, z2), z3)) -> c15(MARK(if(geq(y0, y1), z1, z2))) MARK(div(if(div(minus(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(minus(y0, y1), y2), z1, z2))) MARK(div(if(div(geq(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(geq(y0, y1), y2), z1, z2))) MARK(div(if(div(div(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(div(y0, y1), y2), z1, z2))) MARK(div(if(div(if(y0, y1, y2), y3), z1, z2), z3)) -> c15(MARK(if(div(if(y0, y1, y2), y3), z1, z2))) MARK(div(if(div(s(y0), y1), z1, z2), z3)) -> c15(MARK(if(div(s(y0), y1), z1, z2))) MARK(div(if(div(y0, y1), z1, z2), z3)) -> c15(MARK(if(div(y0, y1), z1, z2))) MARK(div(if(if(minus(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(minus(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(geq(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(geq(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(div(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(div(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(if(y0, y1, y2), y3, y4), z1, z2), z3)) -> c15(MARK(if(if(if(y0, y1, y2), y3, y4), z1, z2))) MARK(div(if(if(s(y0), y1, y2), z1, z2), z3)) -> c15(MARK(if(if(s(y0), y1, y2), z1, z2))) MARK(div(if(if(true, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(true, y0, y1), z1, z2))) MARK(div(if(if(false, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(false, y0, y1), z1, z2))) MARK(div(if(if(y0, y1, y2), z1, z2), z3)) -> c15(MARK(if(if(y0, y1, y2), z1, z2))) MARK(div(if(geq(y0, 0), z1, z2), z3)) -> c15(MARK(if(geq(y0, 0), z1, z2))) MARK(div(if(geq(0, s(y0)), z1, z2), z3)) -> c15(MARK(if(geq(0, s(y0)), z1, z2))) MARK(div(if(s(s(y0)), z1, z2), z3)) -> c15(MARK(if(s(s(y0)), z1, z2))) MARK(div(if(s(div(s(y0), y1)), z1, z2), z3)) -> c15(MARK(if(s(div(s(y0), y1)), z1, z2))) MARK(div(if(s(if(s(y0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(s(y0), y1, y2)), z1, z2))) MARK(div(if(s(if(true, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(true, y0, y1)), z1, z2))) MARK(div(if(s(if(false, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(false, y0, y1)), z1, z2))) MARK(div(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(minus(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(geq(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(div(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2))) MARK(div(if(s(div(div(s(y0), y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(s(y0), y1), y2)), z1, z2))) MARK(div(if(s(div(div(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2), z3)) -> c15(MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2))) MARK(div(if(s(div(if(s(y0), y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2))) MARK(div(if(s(div(if(true, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(true, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(false, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(false, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(y0, y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2))) MARK(div(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(minus(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2))) MARK(div(if(s(if(div(s(y0), y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2), z3)) -> c15(MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2))) MARK(div(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(if(if(true, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(false, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(if(geq(y0, 0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2))) MARK(div(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2))) MARK(div(if(true, minus(y0, y1), z2), z3)) -> c15(MARK(if(true, minus(y0, y1), z2))) MARK(div(if(true, geq(y0, y1), z2), z3)) -> c15(MARK(if(true, geq(y0, y1), z2))) MARK(div(if(true, s(y0), z2), z3)) -> c15(MARK(if(true, s(y0), z2))) MARK(div(if(true, div(s(y0), y1), z2), z3)) -> c15(MARK(if(true, div(s(y0), y1), z2))) MARK(div(if(true, if(s(y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(s(y0), y1, y2), z2))) MARK(div(if(true, if(true, y0, y1), z2), z3)) -> c15(MARK(if(true, if(true, y0, y1), z2))) MARK(div(if(true, if(false, y0, y1), z2), z3)) -> c15(MARK(if(true, if(false, y0, y1), z2))) MARK(div(if(true, div(minus(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(minus(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(minus(0, y0), y1), z2), z3)) -> c15(MARK(if(true, div(minus(0, y0), y1), z2))) MARK(div(if(true, div(minus(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(minus(y0, y1), y2), z2))) MARK(div(if(true, div(geq(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(geq(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(geq(y0, 0), y1), z2), z3)) -> c15(MARK(if(true, div(geq(y0, 0), y1), z2))) MARK(div(if(true, div(geq(0, s(y0)), y1), z2), z3)) -> c15(MARK(if(true, div(geq(0, s(y0)), y1), z2))) MARK(div(if(true, div(geq(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(geq(y0, y1), y2), z2))) MARK(div(if(true, div(div(minus(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(minus(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(geq(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(geq(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(div(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(div(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(if(y0, y1, y2), y3), y4), z2), z3)) -> c15(MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z2))) MARK(div(if(true, div(div(s(y0), y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(s(y0), y1), y2), z2))) MARK(div(if(true, div(div(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(y0, y1), y2), z2))) MARK(div(if(true, div(if(minus(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(geq(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(div(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2), z3)) -> c15(MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2))) MARK(div(if(true, div(if(s(y0), y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(s(y0), y1, y2), y3), z2))) MARK(div(if(true, div(if(true, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(true, y0, y1), y2), z2))) MARK(div(if(true, div(if(false, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(false, y0, y1), y2), z2))) MARK(div(if(true, div(if(y0, y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(y0, y1, y2), y3), z2))) MARK(div(if(true, if(minus(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(minus(0, y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(minus(0, y0), y1, y2), z2))) MARK(div(if(true, if(minus(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(y0, y1), y2, y3), z2))) MARK(div(if(true, if(geq(y0, 0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(y0, 0), y1, y2), z2))) MARK(div(if(true, if(geq(0, s(y0)), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(0, s(y0)), y1, y2), z2))) MARK(div(if(true, if(geq(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(geq(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(y0, y1), y2, y3), z2))) MARK(div(if(true, if(div(minus(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(geq(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(div(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2))) MARK(div(if(true, if(div(s(y0), y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(s(y0), y1), y2, y3), z2))) MARK(div(if(true, if(div(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2), z3)) -> c15(MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2))) MARK(div(if(true, if(if(s(y0), y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z2))) MARK(div(if(true, if(if(true, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(true, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(false, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(false, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(y0, y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(y0, y1, y2), y3, y4), z2))) MARK(div(if(false, z1, minus(y1, y2)), z3)) -> c15(MARK(if(false, z1, minus(y1, y2)))) MARK(div(if(false, z1, geq(y1, y2)), z3)) -> c15(MARK(if(false, z1, geq(y1, y2)))) MARK(div(if(false, z1, s(y1)), z3)) -> c15(MARK(if(false, z1, s(y1)))) MARK(div(if(false, z1, div(s(y1), y2)), z3)) -> c15(MARK(if(false, z1, div(s(y1), y2)))) MARK(div(if(false, z1, if(s(y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(s(y1), y2, y3)))) MARK(div(if(false, z1, if(true, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(true, y1, y2)))) MARK(div(if(false, z1, if(false, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(false, y1, y2)))) MARK(div(if(false, z1, div(minus(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(minus(0, y1), y2)), z3)) -> c15(MARK(if(false, z1, div(minus(0, y1), y2)))) MARK(div(if(false, z1, div(minus(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(y1, y2), y3)))) MARK(div(if(false, z1, div(geq(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(geq(y1, 0), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, 0), y2)))) MARK(div(if(false, z1, div(geq(0, s(y1)), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(0, s(y1)), y2)))) MARK(div(if(false, z1, div(geq(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, y2), y3)))) MARK(div(if(false, z1, div(div(minus(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(minus(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(geq(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(geq(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(div(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(div(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(if(y1, y2, y3), y4), y5)), z3)) -> c15(MARK(if(false, z1, div(div(if(y1, y2, y3), y4), y5)))) MARK(div(if(false, z1, div(div(s(y1), y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(s(y1), y2), y3)))) MARK(div(if(false, z1, div(div(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(y1, y2), y3)))) MARK(div(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(div(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(div(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)), z3)) -> c15(MARK(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)))) MARK(div(if(false, z1, div(if(s(y1), y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(s(y1), y2, y3), y4)))) MARK(div(if(false, z1, div(if(true, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(true, y1, y2), y3)))) MARK(div(if(false, z1, div(if(false, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(false, y1, y2), y3)))) MARK(div(if(false, z1, div(if(y1, y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(y1, y2, y3), y4)))) MARK(div(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(minus(0, y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(minus(0, y1), y2, y3)))) MARK(div(if(false, z1, if(minus(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, 0), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, 0), y2, y3)))) MARK(div(if(false, z1, if(geq(0, s(y1)), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(0, s(y1)), y2, y3)))) MARK(div(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(div(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(div(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)))) MARK(div(if(false, z1, if(div(s(y1), y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(s(y1), y2), y3, y4)))) MARK(div(if(false, z1, if(div(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)), z3)) -> c15(MARK(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)))) MARK(div(if(false, z1, if(if(s(y1), y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(s(y1), y2, y3), y4, y5)))) MARK(div(if(false, z1, if(if(true, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(true, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(false, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(false, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(y1, y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(y1, y2, y3), y4, y5)))) MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(A__IF(a__minus(s(y0), s(y1)), z2, z3), MARK(minus(s(s(y0)), s(s(y1))))) MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10 A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11 S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10 A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11 K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c15_1, c10, c11 ---------------------------------------- (217) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10 A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11 We considered the (Usable) Rules:none And the Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(div(s(y0), z1), z2)) -> c15(MARK(div(s(y0), z1))) MARK(div(div(div(minus(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(minus(y0, y1), y2), z1))) MARK(div(div(div(geq(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(geq(y0, y1), y2), z1))) MARK(div(div(div(div(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(div(y0, y1), y2), z1))) MARK(div(div(div(if(y0, y1, y2), y3), z1), z2)) -> c15(MARK(div(div(if(y0, y1, y2), y3), z1))) MARK(div(div(div(s(y0), y1), z1), z2)) -> c15(MARK(div(div(s(y0), y1), z1))) MARK(div(div(div(y0, y1), z1), z2)) -> c15(MARK(div(div(y0, y1), z1))) MARK(div(div(if(minus(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(minus(y0, y1), y2, y3), z1))) MARK(div(div(if(geq(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(geq(y0, y1), y2, y3), z1))) MARK(div(div(if(div(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(div(y0, y1), y2, y3), z1))) MARK(div(div(if(if(y0, y1, y2), y3, y4), z1), z2)) -> c15(MARK(div(if(if(y0, y1, y2), y3, y4), z1))) MARK(div(div(if(s(y0), y1, y2), z1), z2)) -> c15(MARK(div(if(s(y0), y1, y2), z1))) MARK(div(div(if(true, y0, y1), z1), z2)) -> c15(MARK(div(if(true, y0, y1), z1))) MARK(div(div(if(false, y0, y1), z1), z2)) -> c15(MARK(div(if(false, y0, y1), z1))) MARK(div(div(if(y0, y1, y2), z1), z2)) -> c15(MARK(div(if(y0, y1, y2), z1))) MARK(div(div(minus(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(minus(s(s(y0)), s(s(y1))), z1))) MARK(div(div(geq(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(geq(s(s(y0)), s(s(y1))), z1))) MARK(div(if(minus(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(minus(s(y0), s(y1)), z1, z2))) MARK(div(if(minus(y0, y1), z1, z2), z3)) -> c15(MARK(if(minus(y0, y1), z1, z2))) MARK(div(if(geq(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(geq(s(y0), s(y1)), z1, z2))) MARK(div(if(geq(y0, y1), z1, z2), z3)) -> c15(MARK(if(geq(y0, y1), z1, z2))) MARK(div(if(div(minus(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(minus(y0, y1), y2), z1, z2))) MARK(div(if(div(geq(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(geq(y0, y1), y2), z1, z2))) MARK(div(if(div(div(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(div(y0, y1), y2), z1, z2))) MARK(div(if(div(if(y0, y1, y2), y3), z1, z2), z3)) -> c15(MARK(if(div(if(y0, y1, y2), y3), z1, z2))) MARK(div(if(div(s(y0), y1), z1, z2), z3)) -> c15(MARK(if(div(s(y0), y1), z1, z2))) MARK(div(if(div(y0, y1), z1, z2), z3)) -> c15(MARK(if(div(y0, y1), z1, z2))) MARK(div(if(if(minus(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(minus(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(geq(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(geq(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(div(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(div(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(if(y0, y1, y2), y3, y4), z1, z2), z3)) -> c15(MARK(if(if(if(y0, y1, y2), y3, y4), z1, z2))) MARK(div(if(if(s(y0), y1, y2), z1, z2), z3)) -> c15(MARK(if(if(s(y0), y1, y2), z1, z2))) MARK(div(if(if(true, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(true, y0, y1), z1, z2))) MARK(div(if(if(false, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(false, y0, y1), z1, z2))) MARK(div(if(if(y0, y1, y2), z1, z2), z3)) -> c15(MARK(if(if(y0, y1, y2), z1, z2))) MARK(div(if(geq(y0, 0), z1, z2), z3)) -> c15(MARK(if(geq(y0, 0), z1, z2))) MARK(div(if(geq(0, s(y0)), z1, z2), z3)) -> c15(MARK(if(geq(0, s(y0)), z1, z2))) MARK(div(if(s(s(y0)), z1, z2), z3)) -> c15(MARK(if(s(s(y0)), z1, z2))) MARK(div(if(s(div(s(y0), y1)), z1, z2), z3)) -> c15(MARK(if(s(div(s(y0), y1)), z1, z2))) MARK(div(if(s(if(s(y0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(s(y0), y1, y2)), z1, z2))) MARK(div(if(s(if(true, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(true, y0, y1)), z1, z2))) MARK(div(if(s(if(false, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(false, y0, y1)), z1, z2))) MARK(div(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(minus(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(geq(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(div(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2))) MARK(div(if(s(div(div(s(y0), y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(s(y0), y1), y2)), z1, z2))) MARK(div(if(s(div(div(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2), z3)) -> c15(MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2))) MARK(div(if(s(div(if(s(y0), y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2))) MARK(div(if(s(div(if(true, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(true, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(false, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(false, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(y0, y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2))) MARK(div(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(minus(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2))) MARK(div(if(s(if(div(s(y0), y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2), z3)) -> c15(MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2))) MARK(div(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(if(if(true, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(false, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(if(geq(y0, 0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2))) MARK(div(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2))) MARK(div(if(true, minus(y0, y1), z2), z3)) -> c15(MARK(if(true, minus(y0, y1), z2))) MARK(div(if(true, geq(y0, y1), z2), z3)) -> c15(MARK(if(true, geq(y0, y1), z2))) MARK(div(if(true, s(y0), z2), z3)) -> c15(MARK(if(true, s(y0), z2))) MARK(div(if(true, div(s(y0), y1), z2), z3)) -> c15(MARK(if(true, div(s(y0), y1), z2))) MARK(div(if(true, if(s(y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(s(y0), y1, y2), z2))) MARK(div(if(true, if(true, y0, y1), z2), z3)) -> c15(MARK(if(true, if(true, y0, y1), z2))) MARK(div(if(true, if(false, y0, y1), z2), z3)) -> c15(MARK(if(true, if(false, y0, y1), z2))) MARK(div(if(true, div(minus(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(minus(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(minus(0, y0), y1), z2), z3)) -> c15(MARK(if(true, div(minus(0, y0), y1), z2))) MARK(div(if(true, div(minus(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(minus(y0, y1), y2), z2))) MARK(div(if(true, div(geq(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(geq(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(geq(y0, 0), y1), z2), z3)) -> c15(MARK(if(true, div(geq(y0, 0), y1), z2))) MARK(div(if(true, div(geq(0, s(y0)), y1), z2), z3)) -> c15(MARK(if(true, div(geq(0, s(y0)), y1), z2))) MARK(div(if(true, div(geq(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(geq(y0, y1), y2), z2))) MARK(div(if(true, div(div(minus(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(minus(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(geq(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(geq(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(div(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(div(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(if(y0, y1, y2), y3), y4), z2), z3)) -> c15(MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z2))) MARK(div(if(true, div(div(s(y0), y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(s(y0), y1), y2), z2))) MARK(div(if(true, div(div(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(y0, y1), y2), z2))) MARK(div(if(true, div(if(minus(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(geq(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(div(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2), z3)) -> c15(MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2))) MARK(div(if(true, div(if(s(y0), y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(s(y0), y1, y2), y3), z2))) MARK(div(if(true, div(if(true, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(true, y0, y1), y2), z2))) MARK(div(if(true, div(if(false, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(false, y0, y1), y2), z2))) MARK(div(if(true, div(if(y0, y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(y0, y1, y2), y3), z2))) MARK(div(if(true, if(minus(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(minus(0, y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(minus(0, y0), y1, y2), z2))) MARK(div(if(true, if(minus(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(y0, y1), y2, y3), z2))) MARK(div(if(true, if(geq(y0, 0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(y0, 0), y1, y2), z2))) MARK(div(if(true, if(geq(0, s(y0)), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(0, s(y0)), y1, y2), z2))) MARK(div(if(true, if(geq(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(geq(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(y0, y1), y2, y3), z2))) MARK(div(if(true, if(div(minus(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(geq(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(div(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2))) MARK(div(if(true, if(div(s(y0), y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(s(y0), y1), y2, y3), z2))) MARK(div(if(true, if(div(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2), z3)) -> c15(MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2))) MARK(div(if(true, if(if(s(y0), y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z2))) MARK(div(if(true, if(if(true, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(true, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(false, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(false, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(y0, y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(y0, y1, y2), y3, y4), z2))) MARK(div(if(false, z1, minus(y1, y2)), z3)) -> c15(MARK(if(false, z1, minus(y1, y2)))) MARK(div(if(false, z1, geq(y1, y2)), z3)) -> c15(MARK(if(false, z1, geq(y1, y2)))) MARK(div(if(false, z1, s(y1)), z3)) -> c15(MARK(if(false, z1, s(y1)))) MARK(div(if(false, z1, div(s(y1), y2)), z3)) -> c15(MARK(if(false, z1, div(s(y1), y2)))) MARK(div(if(false, z1, if(s(y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(s(y1), y2, y3)))) MARK(div(if(false, z1, if(true, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(true, y1, y2)))) MARK(div(if(false, z1, if(false, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(false, y1, y2)))) MARK(div(if(false, z1, div(minus(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(minus(0, y1), y2)), z3)) -> c15(MARK(if(false, z1, div(minus(0, y1), y2)))) MARK(div(if(false, z1, div(minus(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(y1, y2), y3)))) MARK(div(if(false, z1, div(geq(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(geq(y1, 0), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, 0), y2)))) MARK(div(if(false, z1, div(geq(0, s(y1)), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(0, s(y1)), y2)))) MARK(div(if(false, z1, div(geq(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, y2), y3)))) MARK(div(if(false, z1, div(div(minus(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(minus(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(geq(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(geq(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(div(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(div(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(if(y1, y2, y3), y4), y5)), z3)) -> c15(MARK(if(false, z1, div(div(if(y1, y2, y3), y4), y5)))) MARK(div(if(false, z1, div(div(s(y1), y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(s(y1), y2), y3)))) MARK(div(if(false, z1, div(div(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(y1, y2), y3)))) MARK(div(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(div(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(div(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)), z3)) -> c15(MARK(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)))) MARK(div(if(false, z1, div(if(s(y1), y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(s(y1), y2, y3), y4)))) MARK(div(if(false, z1, div(if(true, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(true, y1, y2), y3)))) MARK(div(if(false, z1, div(if(false, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(false, y1, y2), y3)))) MARK(div(if(false, z1, div(if(y1, y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(y1, y2, y3), y4)))) MARK(div(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(minus(0, y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(minus(0, y1), y2, y3)))) MARK(div(if(false, z1, if(minus(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, 0), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, 0), y2, y3)))) MARK(div(if(false, z1, if(geq(0, s(y1)), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(0, s(y1)), y2, y3)))) MARK(div(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(div(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(div(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)))) MARK(div(if(false, z1, if(div(s(y1), y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(s(y1), y2), y3, y4)))) MARK(div(if(false, z1, if(div(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)), z3)) -> c15(MARK(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)))) MARK(div(if(false, z1, if(if(s(y1), y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(s(y1), y2, y3), y4, y5)))) MARK(div(if(false, z1, if(if(true, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(true, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(false, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(false, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(y1, y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(y1, y2, y3), y4, y5)))) MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(A__IF(a__minus(s(y0), s(y1)), z2, z3), MARK(minus(s(s(y0)), s(s(y1))))) MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10 A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11 The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(A__DIV(x_1, x_2)) = 0 POL(A__GEQ(x_1, x_2)) = 0 POL(A__IF(x_1, x_2, x_3)) = x_2 + x_3 POL(A__MINUS(x_1, x_2)) = 0 POL(MARK(x_1)) = x_1 POL(a__div(x_1, x_2)) = 0 POL(a__geq(x_1, x_2)) = 0 POL(a__if(x_1, x_2, x_3)) = x_2 + x_3 POL(a__minus(x_1, x_2)) = 0 POL(c1(x_1)) = x_1 POL(c10) = 0 POL(c10(x_1)) = x_1 POL(c11) = 0 POL(c11(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c15(x_1, x_2)) = x_1 + x_2 POL(c16(x_1)) = x_1 POL(c16(x_1, x_2)) = x_1 + x_2 POL(c18(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(div(x_1, x_2)) = x_1 POL(false) = 0 POL(geq(x_1, x_2)) = 0 POL(if(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(mark(x_1)) = x_1 POL(minus(x_1, x_2)) = 0 POL(s(x_1)) = x_1 POL(true) = 0 ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(div(s(y0), z1), z2)) -> c15(MARK(div(s(y0), z1))) MARK(div(div(div(minus(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(minus(y0, y1), y2), z1))) MARK(div(div(div(geq(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(geq(y0, y1), y2), z1))) MARK(div(div(div(div(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(div(y0, y1), y2), z1))) MARK(div(div(div(if(y0, y1, y2), y3), z1), z2)) -> c15(MARK(div(div(if(y0, y1, y2), y3), z1))) MARK(div(div(div(s(y0), y1), z1), z2)) -> c15(MARK(div(div(s(y0), y1), z1))) MARK(div(div(div(y0, y1), z1), z2)) -> c15(MARK(div(div(y0, y1), z1))) MARK(div(div(if(minus(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(minus(y0, y1), y2, y3), z1))) MARK(div(div(if(geq(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(geq(y0, y1), y2, y3), z1))) MARK(div(div(if(div(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(div(y0, y1), y2, y3), z1))) MARK(div(div(if(if(y0, y1, y2), y3, y4), z1), z2)) -> c15(MARK(div(if(if(y0, y1, y2), y3, y4), z1))) MARK(div(div(if(s(y0), y1, y2), z1), z2)) -> c15(MARK(div(if(s(y0), y1, y2), z1))) MARK(div(div(if(true, y0, y1), z1), z2)) -> c15(MARK(div(if(true, y0, y1), z1))) MARK(div(div(if(false, y0, y1), z1), z2)) -> c15(MARK(div(if(false, y0, y1), z1))) MARK(div(div(if(y0, y1, y2), z1), z2)) -> c15(MARK(div(if(y0, y1, y2), z1))) MARK(div(div(minus(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(minus(s(s(y0)), s(s(y1))), z1))) MARK(div(div(geq(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(geq(s(s(y0)), s(s(y1))), z1))) MARK(div(if(minus(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(minus(s(y0), s(y1)), z1, z2))) MARK(div(if(minus(y0, y1), z1, z2), z3)) -> c15(MARK(if(minus(y0, y1), z1, z2))) MARK(div(if(geq(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(geq(s(y0), s(y1)), z1, z2))) MARK(div(if(geq(y0, y1), z1, z2), z3)) -> c15(MARK(if(geq(y0, y1), z1, z2))) MARK(div(if(div(minus(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(minus(y0, y1), y2), z1, z2))) MARK(div(if(div(geq(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(geq(y0, y1), y2), z1, z2))) MARK(div(if(div(div(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(div(y0, y1), y2), z1, z2))) MARK(div(if(div(if(y0, y1, y2), y3), z1, z2), z3)) -> c15(MARK(if(div(if(y0, y1, y2), y3), z1, z2))) MARK(div(if(div(s(y0), y1), z1, z2), z3)) -> c15(MARK(if(div(s(y0), y1), z1, z2))) MARK(div(if(div(y0, y1), z1, z2), z3)) -> c15(MARK(if(div(y0, y1), z1, z2))) MARK(div(if(if(minus(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(minus(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(geq(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(geq(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(div(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(div(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(if(y0, y1, y2), y3, y4), z1, z2), z3)) -> c15(MARK(if(if(if(y0, y1, y2), y3, y4), z1, z2))) MARK(div(if(if(s(y0), y1, y2), z1, z2), z3)) -> c15(MARK(if(if(s(y0), y1, y2), z1, z2))) MARK(div(if(if(true, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(true, y0, y1), z1, z2))) MARK(div(if(if(false, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(false, y0, y1), z1, z2))) MARK(div(if(if(y0, y1, y2), z1, z2), z3)) -> c15(MARK(if(if(y0, y1, y2), z1, z2))) MARK(div(if(geq(y0, 0), z1, z2), z3)) -> c15(MARK(if(geq(y0, 0), z1, z2))) MARK(div(if(geq(0, s(y0)), z1, z2), z3)) -> c15(MARK(if(geq(0, s(y0)), z1, z2))) MARK(div(if(s(s(y0)), z1, z2), z3)) -> c15(MARK(if(s(s(y0)), z1, z2))) MARK(div(if(s(div(s(y0), y1)), z1, z2), z3)) -> c15(MARK(if(s(div(s(y0), y1)), z1, z2))) MARK(div(if(s(if(s(y0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(s(y0), y1, y2)), z1, z2))) MARK(div(if(s(if(true, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(true, y0, y1)), z1, z2))) MARK(div(if(s(if(false, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(false, y0, y1)), z1, z2))) MARK(div(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(minus(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(geq(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(div(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2))) MARK(div(if(s(div(div(s(y0), y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(s(y0), y1), y2)), z1, z2))) MARK(div(if(s(div(div(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2), z3)) -> c15(MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2))) MARK(div(if(s(div(if(s(y0), y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2))) MARK(div(if(s(div(if(true, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(true, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(false, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(false, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(y0, y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2))) MARK(div(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(minus(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2))) MARK(div(if(s(if(div(s(y0), y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2), z3)) -> c15(MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2))) MARK(div(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(if(if(true, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(false, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(if(geq(y0, 0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2))) MARK(div(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2))) MARK(div(if(true, minus(y0, y1), z2), z3)) -> c15(MARK(if(true, minus(y0, y1), z2))) MARK(div(if(true, geq(y0, y1), z2), z3)) -> c15(MARK(if(true, geq(y0, y1), z2))) MARK(div(if(true, s(y0), z2), z3)) -> c15(MARK(if(true, s(y0), z2))) MARK(div(if(true, div(s(y0), y1), z2), z3)) -> c15(MARK(if(true, div(s(y0), y1), z2))) MARK(div(if(true, if(s(y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(s(y0), y1, y2), z2))) MARK(div(if(true, if(true, y0, y1), z2), z3)) -> c15(MARK(if(true, if(true, y0, y1), z2))) MARK(div(if(true, if(false, y0, y1), z2), z3)) -> c15(MARK(if(true, if(false, y0, y1), z2))) MARK(div(if(true, div(minus(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(minus(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(minus(0, y0), y1), z2), z3)) -> c15(MARK(if(true, div(minus(0, y0), y1), z2))) MARK(div(if(true, div(minus(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(minus(y0, y1), y2), z2))) MARK(div(if(true, div(geq(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(geq(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(geq(y0, 0), y1), z2), z3)) -> c15(MARK(if(true, div(geq(y0, 0), y1), z2))) MARK(div(if(true, div(geq(0, s(y0)), y1), z2), z3)) -> c15(MARK(if(true, div(geq(0, s(y0)), y1), z2))) MARK(div(if(true, div(geq(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(geq(y0, y1), y2), z2))) MARK(div(if(true, div(div(minus(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(minus(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(geq(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(geq(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(div(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(div(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(if(y0, y1, y2), y3), y4), z2), z3)) -> c15(MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z2))) MARK(div(if(true, div(div(s(y0), y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(s(y0), y1), y2), z2))) MARK(div(if(true, div(div(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(y0, y1), y2), z2))) MARK(div(if(true, div(if(minus(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(geq(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(div(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2), z3)) -> c15(MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2))) MARK(div(if(true, div(if(s(y0), y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(s(y0), y1, y2), y3), z2))) MARK(div(if(true, div(if(true, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(true, y0, y1), y2), z2))) MARK(div(if(true, div(if(false, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(false, y0, y1), y2), z2))) MARK(div(if(true, div(if(y0, y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(y0, y1, y2), y3), z2))) MARK(div(if(true, if(minus(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(minus(0, y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(minus(0, y0), y1, y2), z2))) MARK(div(if(true, if(minus(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(y0, y1), y2, y3), z2))) MARK(div(if(true, if(geq(y0, 0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(y0, 0), y1, y2), z2))) MARK(div(if(true, if(geq(0, s(y0)), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(0, s(y0)), y1, y2), z2))) MARK(div(if(true, if(geq(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(geq(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(y0, y1), y2, y3), z2))) MARK(div(if(true, if(div(minus(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(geq(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(div(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2))) MARK(div(if(true, if(div(s(y0), y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(s(y0), y1), y2, y3), z2))) MARK(div(if(true, if(div(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2), z3)) -> c15(MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2))) MARK(div(if(true, if(if(s(y0), y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z2))) MARK(div(if(true, if(if(true, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(true, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(false, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(false, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(y0, y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(y0, y1, y2), y3, y4), z2))) MARK(div(if(false, z1, minus(y1, y2)), z3)) -> c15(MARK(if(false, z1, minus(y1, y2)))) MARK(div(if(false, z1, geq(y1, y2)), z3)) -> c15(MARK(if(false, z1, geq(y1, y2)))) MARK(div(if(false, z1, s(y1)), z3)) -> c15(MARK(if(false, z1, s(y1)))) MARK(div(if(false, z1, div(s(y1), y2)), z3)) -> c15(MARK(if(false, z1, div(s(y1), y2)))) MARK(div(if(false, z1, if(s(y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(s(y1), y2, y3)))) MARK(div(if(false, z1, if(true, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(true, y1, y2)))) MARK(div(if(false, z1, if(false, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(false, y1, y2)))) MARK(div(if(false, z1, div(minus(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(minus(0, y1), y2)), z3)) -> c15(MARK(if(false, z1, div(minus(0, y1), y2)))) MARK(div(if(false, z1, div(minus(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(y1, y2), y3)))) MARK(div(if(false, z1, div(geq(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(geq(y1, 0), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, 0), y2)))) MARK(div(if(false, z1, div(geq(0, s(y1)), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(0, s(y1)), y2)))) MARK(div(if(false, z1, div(geq(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, y2), y3)))) MARK(div(if(false, z1, div(div(minus(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(minus(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(geq(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(geq(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(div(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(div(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(if(y1, y2, y3), y4), y5)), z3)) -> c15(MARK(if(false, z1, div(div(if(y1, y2, y3), y4), y5)))) MARK(div(if(false, z1, div(div(s(y1), y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(s(y1), y2), y3)))) MARK(div(if(false, z1, div(div(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(y1, y2), y3)))) MARK(div(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(div(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(div(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)), z3)) -> c15(MARK(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)))) MARK(div(if(false, z1, div(if(s(y1), y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(s(y1), y2, y3), y4)))) MARK(div(if(false, z1, div(if(true, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(true, y1, y2), y3)))) MARK(div(if(false, z1, div(if(false, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(false, y1, y2), y3)))) MARK(div(if(false, z1, div(if(y1, y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(y1, y2, y3), y4)))) MARK(div(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(minus(0, y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(minus(0, y1), y2, y3)))) MARK(div(if(false, z1, if(minus(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, 0), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, 0), y2, y3)))) MARK(div(if(false, z1, if(geq(0, s(y1)), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(0, s(y1)), y2, y3)))) MARK(div(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(div(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(div(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)))) MARK(div(if(false, z1, if(div(s(y1), y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(s(y1), y2), y3, y4)))) MARK(div(if(false, z1, if(div(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)), z3)) -> c15(MARK(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)))) MARK(div(if(false, z1, if(if(s(y1), y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(s(y1), y2, y3), y4, y5)))) MARK(div(if(false, z1, if(if(true, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(true, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(false, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(false, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(y1, y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(y1, y2, y3), y4, y5)))) MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(A__IF(a__minus(s(y0), s(y1)), z2, z3), MARK(minus(s(s(y0)), s(s(y1))))) MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10 A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11 S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10 A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11 Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c15_1, c10, c11 ---------------------------------------- (219) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(if(geq(z0, z1), x2, x3)) -> c16(MARK(geq(z0, z1))) by MARK(if(geq(s(s(y0)), s(s(y1))), z2, z3)) -> c16(MARK(geq(s(s(y0)), s(s(y1))))) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(div(s(y0), z1), z2)) -> c15(MARK(div(s(y0), z1))) MARK(div(div(div(minus(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(minus(y0, y1), y2), z1))) MARK(div(div(div(geq(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(geq(y0, y1), y2), z1))) MARK(div(div(div(div(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(div(y0, y1), y2), z1))) MARK(div(div(div(if(y0, y1, y2), y3), z1), z2)) -> c15(MARK(div(div(if(y0, y1, y2), y3), z1))) MARK(div(div(div(s(y0), y1), z1), z2)) -> c15(MARK(div(div(s(y0), y1), z1))) MARK(div(div(div(y0, y1), z1), z2)) -> c15(MARK(div(div(y0, y1), z1))) MARK(div(div(if(minus(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(minus(y0, y1), y2, y3), z1))) MARK(div(div(if(geq(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(geq(y0, y1), y2, y3), z1))) MARK(div(div(if(div(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(div(y0, y1), y2, y3), z1))) MARK(div(div(if(if(y0, y1, y2), y3, y4), z1), z2)) -> c15(MARK(div(if(if(y0, y1, y2), y3, y4), z1))) MARK(div(div(if(s(y0), y1, y2), z1), z2)) -> c15(MARK(div(if(s(y0), y1, y2), z1))) MARK(div(div(if(true, y0, y1), z1), z2)) -> c15(MARK(div(if(true, y0, y1), z1))) MARK(div(div(if(false, y0, y1), z1), z2)) -> c15(MARK(div(if(false, y0, y1), z1))) MARK(div(div(if(y0, y1, y2), z1), z2)) -> c15(MARK(div(if(y0, y1, y2), z1))) MARK(div(div(minus(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(minus(s(s(y0)), s(s(y1))), z1))) MARK(div(div(geq(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(geq(s(s(y0)), s(s(y1))), z1))) MARK(div(if(minus(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(minus(s(y0), s(y1)), z1, z2))) MARK(div(if(minus(y0, y1), z1, z2), z3)) -> c15(MARK(if(minus(y0, y1), z1, z2))) MARK(div(if(geq(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(geq(s(y0), s(y1)), z1, z2))) MARK(div(if(geq(y0, y1), z1, z2), z3)) -> c15(MARK(if(geq(y0, y1), z1, z2))) MARK(div(if(div(minus(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(minus(y0, y1), y2), z1, z2))) MARK(div(if(div(geq(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(geq(y0, y1), y2), z1, z2))) MARK(div(if(div(div(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(div(y0, y1), y2), z1, z2))) MARK(div(if(div(if(y0, y1, y2), y3), z1, z2), z3)) -> c15(MARK(if(div(if(y0, y1, y2), y3), z1, z2))) MARK(div(if(div(s(y0), y1), z1, z2), z3)) -> c15(MARK(if(div(s(y0), y1), z1, z2))) MARK(div(if(div(y0, y1), z1, z2), z3)) -> c15(MARK(if(div(y0, y1), z1, z2))) MARK(div(if(if(minus(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(minus(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(geq(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(geq(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(div(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(div(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(if(y0, y1, y2), y3, y4), z1, z2), z3)) -> c15(MARK(if(if(if(y0, y1, y2), y3, y4), z1, z2))) MARK(div(if(if(s(y0), y1, y2), z1, z2), z3)) -> c15(MARK(if(if(s(y0), y1, y2), z1, z2))) MARK(div(if(if(true, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(true, y0, y1), z1, z2))) MARK(div(if(if(false, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(false, y0, y1), z1, z2))) MARK(div(if(if(y0, y1, y2), z1, z2), z3)) -> c15(MARK(if(if(y0, y1, y2), z1, z2))) MARK(div(if(geq(y0, 0), z1, z2), z3)) -> c15(MARK(if(geq(y0, 0), z1, z2))) MARK(div(if(geq(0, s(y0)), z1, z2), z3)) -> c15(MARK(if(geq(0, s(y0)), z1, z2))) MARK(div(if(s(s(y0)), z1, z2), z3)) -> c15(MARK(if(s(s(y0)), z1, z2))) MARK(div(if(s(div(s(y0), y1)), z1, z2), z3)) -> c15(MARK(if(s(div(s(y0), y1)), z1, z2))) MARK(div(if(s(if(s(y0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(s(y0), y1, y2)), z1, z2))) MARK(div(if(s(if(true, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(true, y0, y1)), z1, z2))) MARK(div(if(s(if(false, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(false, y0, y1)), z1, z2))) MARK(div(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(minus(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(geq(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(div(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2))) MARK(div(if(s(div(div(s(y0), y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(s(y0), y1), y2)), z1, z2))) MARK(div(if(s(div(div(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2), z3)) -> c15(MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2))) MARK(div(if(s(div(if(s(y0), y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2))) MARK(div(if(s(div(if(true, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(true, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(false, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(false, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(y0, y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2))) MARK(div(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(minus(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2))) MARK(div(if(s(if(div(s(y0), y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2), z3)) -> c15(MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2))) MARK(div(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(if(if(true, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(false, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(if(geq(y0, 0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2))) MARK(div(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2))) MARK(div(if(true, minus(y0, y1), z2), z3)) -> c15(MARK(if(true, minus(y0, y1), z2))) MARK(div(if(true, geq(y0, y1), z2), z3)) -> c15(MARK(if(true, geq(y0, y1), z2))) MARK(div(if(true, s(y0), z2), z3)) -> c15(MARK(if(true, s(y0), z2))) MARK(div(if(true, div(s(y0), y1), z2), z3)) -> c15(MARK(if(true, div(s(y0), y1), z2))) MARK(div(if(true, if(s(y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(s(y0), y1, y2), z2))) MARK(div(if(true, if(true, y0, y1), z2), z3)) -> c15(MARK(if(true, if(true, y0, y1), z2))) MARK(div(if(true, if(false, y0, y1), z2), z3)) -> c15(MARK(if(true, if(false, y0, y1), z2))) MARK(div(if(true, div(minus(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(minus(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(minus(0, y0), y1), z2), z3)) -> c15(MARK(if(true, div(minus(0, y0), y1), z2))) MARK(div(if(true, div(minus(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(minus(y0, y1), y2), z2))) MARK(div(if(true, div(geq(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(geq(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(geq(y0, 0), y1), z2), z3)) -> c15(MARK(if(true, div(geq(y0, 0), y1), z2))) MARK(div(if(true, div(geq(0, s(y0)), y1), z2), z3)) -> c15(MARK(if(true, div(geq(0, s(y0)), y1), z2))) MARK(div(if(true, div(geq(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(geq(y0, y1), y2), z2))) MARK(div(if(true, div(div(minus(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(minus(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(geq(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(geq(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(div(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(div(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(if(y0, y1, y2), y3), y4), z2), z3)) -> c15(MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z2))) MARK(div(if(true, div(div(s(y0), y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(s(y0), y1), y2), z2))) MARK(div(if(true, div(div(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(y0, y1), y2), z2))) MARK(div(if(true, div(if(minus(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(geq(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(div(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2), z3)) -> c15(MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2))) MARK(div(if(true, div(if(s(y0), y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(s(y0), y1, y2), y3), z2))) MARK(div(if(true, div(if(true, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(true, y0, y1), y2), z2))) MARK(div(if(true, div(if(false, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(false, y0, y1), y2), z2))) MARK(div(if(true, div(if(y0, y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(y0, y1, y2), y3), z2))) MARK(div(if(true, if(minus(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(minus(0, y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(minus(0, y0), y1, y2), z2))) MARK(div(if(true, if(minus(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(y0, y1), y2, y3), z2))) MARK(div(if(true, if(geq(y0, 0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(y0, 0), y1, y2), z2))) MARK(div(if(true, if(geq(0, s(y0)), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(0, s(y0)), y1, y2), z2))) MARK(div(if(true, if(geq(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(geq(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(y0, y1), y2, y3), z2))) MARK(div(if(true, if(div(minus(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(geq(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(div(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2))) MARK(div(if(true, if(div(s(y0), y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(s(y0), y1), y2, y3), z2))) MARK(div(if(true, if(div(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2), z3)) -> c15(MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2))) MARK(div(if(true, if(if(s(y0), y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z2))) MARK(div(if(true, if(if(true, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(true, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(false, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(false, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(y0, y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(y0, y1, y2), y3, y4), z2))) MARK(div(if(false, z1, minus(y1, y2)), z3)) -> c15(MARK(if(false, z1, minus(y1, y2)))) MARK(div(if(false, z1, geq(y1, y2)), z3)) -> c15(MARK(if(false, z1, geq(y1, y2)))) MARK(div(if(false, z1, s(y1)), z3)) -> c15(MARK(if(false, z1, s(y1)))) MARK(div(if(false, z1, div(s(y1), y2)), z3)) -> c15(MARK(if(false, z1, div(s(y1), y2)))) MARK(div(if(false, z1, if(s(y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(s(y1), y2, y3)))) MARK(div(if(false, z1, if(true, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(true, y1, y2)))) MARK(div(if(false, z1, if(false, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(false, y1, y2)))) MARK(div(if(false, z1, div(minus(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(minus(0, y1), y2)), z3)) -> c15(MARK(if(false, z1, div(minus(0, y1), y2)))) MARK(div(if(false, z1, div(minus(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(y1, y2), y3)))) MARK(div(if(false, z1, div(geq(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(geq(y1, 0), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, 0), y2)))) MARK(div(if(false, z1, div(geq(0, s(y1)), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(0, s(y1)), y2)))) MARK(div(if(false, z1, div(geq(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, y2), y3)))) MARK(div(if(false, z1, div(div(minus(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(minus(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(geq(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(geq(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(div(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(div(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(if(y1, y2, y3), y4), y5)), z3)) -> c15(MARK(if(false, z1, div(div(if(y1, y2, y3), y4), y5)))) MARK(div(if(false, z1, div(div(s(y1), y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(s(y1), y2), y3)))) MARK(div(if(false, z1, div(div(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(y1, y2), y3)))) MARK(div(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(div(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(div(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)), z3)) -> c15(MARK(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)))) MARK(div(if(false, z1, div(if(s(y1), y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(s(y1), y2, y3), y4)))) MARK(div(if(false, z1, div(if(true, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(true, y1, y2), y3)))) MARK(div(if(false, z1, div(if(false, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(false, y1, y2), y3)))) MARK(div(if(false, z1, div(if(y1, y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(y1, y2, y3), y4)))) MARK(div(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(minus(0, y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(minus(0, y1), y2, y3)))) MARK(div(if(false, z1, if(minus(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, 0), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, 0), y2, y3)))) MARK(div(if(false, z1, if(geq(0, s(y1)), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(0, s(y1)), y2, y3)))) MARK(div(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(div(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(div(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)))) MARK(div(if(false, z1, if(div(s(y1), y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(s(y1), y2), y3, y4)))) MARK(div(if(false, z1, if(div(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)), z3)) -> c15(MARK(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)))) MARK(div(if(false, z1, if(if(s(y1), y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(s(y1), y2, y3), y4, y5)))) MARK(div(if(false, z1, if(if(true, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(true, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(false, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(false, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(y1, y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(y1, y2, y3), y4, y5)))) MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(A__IF(a__minus(s(y0), s(y1)), z2, z3), MARK(minus(s(s(y0)), s(s(y1))))) MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10 A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11 MARK(if(geq(s(s(y0)), s(s(y1))), z2, z3)) -> c16(MARK(geq(s(s(y0)), s(s(y1))))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10 A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11 Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c15_1, c10, c11 ---------------------------------------- (221) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: A__IF(false, z0, if(minus(0, y0), y1, y2)) -> c11 A__IF(true, if(minus(0, y0), y1, y2), z1) -> c10 ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(div(s(y0), z1), z2)) -> c15(MARK(div(s(y0), z1))) MARK(div(div(div(minus(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(minus(y0, y1), y2), z1))) MARK(div(div(div(geq(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(geq(y0, y1), y2), z1))) MARK(div(div(div(div(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(div(y0, y1), y2), z1))) MARK(div(div(div(if(y0, y1, y2), y3), z1), z2)) -> c15(MARK(div(div(if(y0, y1, y2), y3), z1))) MARK(div(div(div(s(y0), y1), z1), z2)) -> c15(MARK(div(div(s(y0), y1), z1))) MARK(div(div(div(y0, y1), z1), z2)) -> c15(MARK(div(div(y0, y1), z1))) MARK(div(div(if(minus(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(minus(y0, y1), y2, y3), z1))) MARK(div(div(if(geq(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(geq(y0, y1), y2, y3), z1))) MARK(div(div(if(div(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(div(y0, y1), y2, y3), z1))) MARK(div(div(if(if(y0, y1, y2), y3, y4), z1), z2)) -> c15(MARK(div(if(if(y0, y1, y2), y3, y4), z1))) MARK(div(div(if(s(y0), y1, y2), z1), z2)) -> c15(MARK(div(if(s(y0), y1, y2), z1))) MARK(div(div(if(true, y0, y1), z1), z2)) -> c15(MARK(div(if(true, y0, y1), z1))) MARK(div(div(if(false, y0, y1), z1), z2)) -> c15(MARK(div(if(false, y0, y1), z1))) MARK(div(div(if(y0, y1, y2), z1), z2)) -> c15(MARK(div(if(y0, y1, y2), z1))) MARK(div(div(minus(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(minus(s(s(y0)), s(s(y1))), z1))) MARK(div(div(geq(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(geq(s(s(y0)), s(s(y1))), z1))) MARK(div(if(minus(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(minus(s(y0), s(y1)), z1, z2))) MARK(div(if(minus(y0, y1), z1, z2), z3)) -> c15(MARK(if(minus(y0, y1), z1, z2))) MARK(div(if(geq(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(geq(s(y0), s(y1)), z1, z2))) MARK(div(if(geq(y0, y1), z1, z2), z3)) -> c15(MARK(if(geq(y0, y1), z1, z2))) MARK(div(if(div(minus(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(minus(y0, y1), y2), z1, z2))) MARK(div(if(div(geq(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(geq(y0, y1), y2), z1, z2))) MARK(div(if(div(div(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(div(y0, y1), y2), z1, z2))) MARK(div(if(div(if(y0, y1, y2), y3), z1, z2), z3)) -> c15(MARK(if(div(if(y0, y1, y2), y3), z1, z2))) MARK(div(if(div(s(y0), y1), z1, z2), z3)) -> c15(MARK(if(div(s(y0), y1), z1, z2))) MARK(div(if(div(y0, y1), z1, z2), z3)) -> c15(MARK(if(div(y0, y1), z1, z2))) MARK(div(if(if(minus(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(minus(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(geq(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(geq(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(div(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(div(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(if(y0, y1, y2), y3, y4), z1, z2), z3)) -> c15(MARK(if(if(if(y0, y1, y2), y3, y4), z1, z2))) MARK(div(if(if(s(y0), y1, y2), z1, z2), z3)) -> c15(MARK(if(if(s(y0), y1, y2), z1, z2))) MARK(div(if(if(true, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(true, y0, y1), z1, z2))) MARK(div(if(if(false, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(false, y0, y1), z1, z2))) MARK(div(if(if(y0, y1, y2), z1, z2), z3)) -> c15(MARK(if(if(y0, y1, y2), z1, z2))) MARK(div(if(geq(y0, 0), z1, z2), z3)) -> c15(MARK(if(geq(y0, 0), z1, z2))) MARK(div(if(geq(0, s(y0)), z1, z2), z3)) -> c15(MARK(if(geq(0, s(y0)), z1, z2))) MARK(div(if(s(s(y0)), z1, z2), z3)) -> c15(MARK(if(s(s(y0)), z1, z2))) MARK(div(if(s(div(s(y0), y1)), z1, z2), z3)) -> c15(MARK(if(s(div(s(y0), y1)), z1, z2))) MARK(div(if(s(if(s(y0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(s(y0), y1, y2)), z1, z2))) MARK(div(if(s(if(true, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(true, y0, y1)), z1, z2))) MARK(div(if(s(if(false, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(false, y0, y1)), z1, z2))) MARK(div(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(minus(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(geq(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(div(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2))) MARK(div(if(s(div(div(s(y0), y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(s(y0), y1), y2)), z1, z2))) MARK(div(if(s(div(div(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2), z3)) -> c15(MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2))) MARK(div(if(s(div(if(s(y0), y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2))) MARK(div(if(s(div(if(true, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(true, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(false, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(false, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(y0, y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2))) MARK(div(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(minus(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2))) MARK(div(if(s(if(div(s(y0), y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2), z3)) -> c15(MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2))) MARK(div(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(if(if(true, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(false, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(if(geq(y0, 0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2))) MARK(div(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2))) MARK(div(if(true, minus(y0, y1), z2), z3)) -> c15(MARK(if(true, minus(y0, y1), z2))) MARK(div(if(true, geq(y0, y1), z2), z3)) -> c15(MARK(if(true, geq(y0, y1), z2))) MARK(div(if(true, s(y0), z2), z3)) -> c15(MARK(if(true, s(y0), z2))) MARK(div(if(true, div(s(y0), y1), z2), z3)) -> c15(MARK(if(true, div(s(y0), y1), z2))) MARK(div(if(true, if(s(y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(s(y0), y1, y2), z2))) MARK(div(if(true, if(true, y0, y1), z2), z3)) -> c15(MARK(if(true, if(true, y0, y1), z2))) MARK(div(if(true, if(false, y0, y1), z2), z3)) -> c15(MARK(if(true, if(false, y0, y1), z2))) MARK(div(if(true, div(minus(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(minus(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(minus(0, y0), y1), z2), z3)) -> c15(MARK(if(true, div(minus(0, y0), y1), z2))) MARK(div(if(true, div(minus(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(minus(y0, y1), y2), z2))) MARK(div(if(true, div(geq(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(geq(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(geq(y0, 0), y1), z2), z3)) -> c15(MARK(if(true, div(geq(y0, 0), y1), z2))) MARK(div(if(true, div(geq(0, s(y0)), y1), z2), z3)) -> c15(MARK(if(true, div(geq(0, s(y0)), y1), z2))) MARK(div(if(true, div(geq(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(geq(y0, y1), y2), z2))) MARK(div(if(true, div(div(minus(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(minus(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(geq(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(geq(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(div(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(div(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(if(y0, y1, y2), y3), y4), z2), z3)) -> c15(MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z2))) MARK(div(if(true, div(div(s(y0), y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(s(y0), y1), y2), z2))) MARK(div(if(true, div(div(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(y0, y1), y2), z2))) MARK(div(if(true, div(if(minus(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(geq(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(div(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2), z3)) -> c15(MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2))) MARK(div(if(true, div(if(s(y0), y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(s(y0), y1, y2), y3), z2))) MARK(div(if(true, div(if(true, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(true, y0, y1), y2), z2))) MARK(div(if(true, div(if(false, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(false, y0, y1), y2), z2))) MARK(div(if(true, div(if(y0, y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(y0, y1, y2), y3), z2))) MARK(div(if(true, if(minus(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(minus(0, y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(minus(0, y0), y1, y2), z2))) MARK(div(if(true, if(minus(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(y0, y1), y2, y3), z2))) MARK(div(if(true, if(geq(y0, 0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(y0, 0), y1, y2), z2))) MARK(div(if(true, if(geq(0, s(y0)), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(0, s(y0)), y1, y2), z2))) MARK(div(if(true, if(geq(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(geq(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(y0, y1), y2, y3), z2))) MARK(div(if(true, if(div(minus(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(geq(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(div(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2))) MARK(div(if(true, if(div(s(y0), y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(s(y0), y1), y2, y3), z2))) MARK(div(if(true, if(div(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2), z3)) -> c15(MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2))) MARK(div(if(true, if(if(s(y0), y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z2))) MARK(div(if(true, if(if(true, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(true, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(false, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(false, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(y0, y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(y0, y1, y2), y3, y4), z2))) MARK(div(if(false, z1, minus(y1, y2)), z3)) -> c15(MARK(if(false, z1, minus(y1, y2)))) MARK(div(if(false, z1, geq(y1, y2)), z3)) -> c15(MARK(if(false, z1, geq(y1, y2)))) MARK(div(if(false, z1, s(y1)), z3)) -> c15(MARK(if(false, z1, s(y1)))) MARK(div(if(false, z1, div(s(y1), y2)), z3)) -> c15(MARK(if(false, z1, div(s(y1), y2)))) MARK(div(if(false, z1, if(s(y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(s(y1), y2, y3)))) MARK(div(if(false, z1, if(true, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(true, y1, y2)))) MARK(div(if(false, z1, if(false, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(false, y1, y2)))) MARK(div(if(false, z1, div(minus(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(minus(0, y1), y2)), z3)) -> c15(MARK(if(false, z1, div(minus(0, y1), y2)))) MARK(div(if(false, z1, div(minus(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(y1, y2), y3)))) MARK(div(if(false, z1, div(geq(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(geq(y1, 0), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, 0), y2)))) MARK(div(if(false, z1, div(geq(0, s(y1)), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(0, s(y1)), y2)))) MARK(div(if(false, z1, div(geq(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, y2), y3)))) MARK(div(if(false, z1, div(div(minus(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(minus(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(geq(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(geq(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(div(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(div(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(if(y1, y2, y3), y4), y5)), z3)) -> c15(MARK(if(false, z1, div(div(if(y1, y2, y3), y4), y5)))) MARK(div(if(false, z1, div(div(s(y1), y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(s(y1), y2), y3)))) MARK(div(if(false, z1, div(div(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(y1, y2), y3)))) MARK(div(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(div(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(div(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)), z3)) -> c15(MARK(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)))) MARK(div(if(false, z1, div(if(s(y1), y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(s(y1), y2, y3), y4)))) MARK(div(if(false, z1, div(if(true, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(true, y1, y2), y3)))) MARK(div(if(false, z1, div(if(false, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(false, y1, y2), y3)))) MARK(div(if(false, z1, div(if(y1, y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(y1, y2, y3), y4)))) MARK(div(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(minus(0, y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(minus(0, y1), y2, y3)))) MARK(div(if(false, z1, if(minus(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, 0), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, 0), y2, y3)))) MARK(div(if(false, z1, if(geq(0, s(y1)), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(0, s(y1)), y2, y3)))) MARK(div(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(div(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(div(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)))) MARK(div(if(false, z1, if(div(s(y1), y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(s(y1), y2), y3, y4)))) MARK(div(if(false, z1, if(div(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)), z3)) -> c15(MARK(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)))) MARK(div(if(false, z1, if(if(s(y1), y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(s(y1), y2, y3), y4, y5)))) MARK(div(if(false, z1, if(if(true, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(true, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(false, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(false, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(y1, y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(y1, y2, y3), y4, y5)))) MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(A__IF(a__minus(s(y0), s(y1)), z2, z3), MARK(minus(s(s(y0)), s(s(y1))))) MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(MARK(minus(s(s(y0)), s(s(y1))))) MARK(if(geq(s(s(y0)), s(s(y1))), z2, z3)) -> c16(MARK(geq(s(s(y0)), s(s(y1))))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c15_1 ---------------------------------------- (223) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(if(div(x0, z1), x2, x3)) -> c16(MARK(div(x0, z1))) by MARK(if(div(s(y0), z1), z2, z3)) -> c16(MARK(div(s(y0), z1))) MARK(if(div(div(minus(y0, y1), y2), z1), z2, z3)) -> c16(MARK(div(div(minus(y0, y1), y2), z1))) MARK(if(div(div(geq(y0, y1), y2), z1), z2, z3)) -> c16(MARK(div(div(geq(y0, y1), y2), z1))) MARK(if(div(div(div(y0, y1), y2), z1), z2, z3)) -> c16(MARK(div(div(div(y0, y1), y2), z1))) MARK(if(div(div(if(y0, y1, y2), y3), z1), z2, z3)) -> c16(MARK(div(div(if(y0, y1, y2), y3), z1))) MARK(if(div(div(s(y0), y1), z1), z2, z3)) -> c16(MARK(div(div(s(y0), y1), z1))) MARK(if(div(if(minus(y0, y1), y2, y3), z1), z2, z3)) -> c16(MARK(div(if(minus(y0, y1), y2, y3), z1))) MARK(if(div(if(geq(y0, y1), y2, y3), z1), z2, z3)) -> c16(MARK(div(if(geq(y0, y1), y2, y3), z1))) MARK(if(div(if(div(y0, y1), y2, y3), z1), z2, z3)) -> c16(MARK(div(if(div(y0, y1), y2, y3), z1))) MARK(if(div(if(if(y0, y1, y2), y3, y4), z1), z2, z3)) -> c16(MARK(div(if(if(y0, y1, y2), y3, y4), z1))) MARK(if(div(if(s(y0), y1, y2), z1), z2, z3)) -> c16(MARK(div(if(s(y0), y1, y2), z1))) MARK(if(div(if(true, y0, y1), z1), z2, z3)) -> c16(MARK(div(if(true, y0, y1), z1))) MARK(if(div(if(false, y0, y1), z1), z2, z3)) -> c16(MARK(div(if(false, y0, y1), z1))) MARK(if(div(minus(s(s(y0)), s(s(y1))), z1), z2, z3)) -> c16(MARK(div(minus(s(s(y0)), s(s(y1))), z1))) MARK(if(div(geq(s(s(y0)), s(s(y1))), z1), z2, z3)) -> c16(MARK(div(geq(s(s(y0)), s(s(y1))), z1))) MARK(if(div(div(div(minus(y0, y1), y2), y3), z1), z2, z3)) -> c16(MARK(div(div(div(minus(y0, y1), y2), y3), z1))) MARK(if(div(div(div(geq(y0, y1), y2), y3), z1), z2, z3)) -> c16(MARK(div(div(div(geq(y0, y1), y2), y3), z1))) MARK(if(div(div(div(div(y0, y1), y2), y3), z1), z2, z3)) -> c16(MARK(div(div(div(div(y0, y1), y2), y3), z1))) MARK(if(div(div(div(if(y0, y1, y2), y3), y4), z1), z2, z3)) -> c16(MARK(div(div(div(if(y0, y1, y2), y3), y4), z1))) MARK(if(div(div(div(s(y0), y1), y2), z1), z2, z3)) -> c16(MARK(div(div(div(s(y0), y1), y2), z1))) MARK(if(div(div(if(minus(y0, y1), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(div(if(minus(y0, y1), y2, y3), y4), z1))) MARK(if(div(div(if(geq(y0, y1), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(div(if(geq(y0, y1), y2, y3), y4), z1))) MARK(if(div(div(if(div(y0, y1), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(div(if(div(y0, y1), y2, y3), y4), z1))) MARK(if(div(div(if(if(y0, y1, y2), y3, y4), y5), z1), z2, z3)) -> c16(MARK(div(div(if(if(y0, y1, y2), y3, y4), y5), z1))) MARK(if(div(div(if(s(y0), y1, y2), y3), z1), z2, z3)) -> c16(MARK(div(div(if(s(y0), y1, y2), y3), z1))) MARK(if(div(div(if(true, y0, y1), y2), z1), z2, z3)) -> c16(MARK(div(div(if(true, y0, y1), y2), z1))) MARK(if(div(div(if(false, y0, y1), y2), z1), z2, z3)) -> c16(MARK(div(div(if(false, y0, y1), y2), z1))) MARK(if(div(div(minus(s(s(y0)), s(s(y1))), y2), z1), z2, z3)) -> c16(MARK(div(div(minus(s(s(y0)), s(s(y1))), y2), z1))) MARK(if(div(div(geq(s(s(y0)), s(s(y1))), y2), z1), z2, z3)) -> c16(MARK(div(div(geq(s(s(y0)), s(s(y1))), y2), z1))) MARK(if(div(if(minus(s(y0), s(y1)), y2, y3), z1), z2, z3)) -> c16(MARK(div(if(minus(s(y0), s(y1)), y2, y3), z1))) MARK(if(div(if(geq(s(y0), s(y1)), y2, y3), z1), z2, z3)) -> c16(MARK(div(if(geq(s(y0), s(y1)), y2, y3), z1))) MARK(if(div(if(div(minus(y0, y1), y2), y3, y4), z1), z2, z3)) -> c16(MARK(div(if(div(minus(y0, y1), y2), y3, y4), z1))) MARK(if(div(if(div(geq(y0, y1), y2), y3, y4), z1), z2, z3)) -> c16(MARK(div(if(div(geq(y0, y1), y2), y3, y4), z1))) MARK(if(div(if(div(div(y0, y1), y2), y3, y4), z1), z2, z3)) -> c16(MARK(div(if(div(div(y0, y1), y2), y3, y4), z1))) MARK(if(div(if(div(if(y0, y1, y2), y3), y4, y5), z1), z2, z3)) -> 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div(if(geq(y0, y1), y2, y3), y4), y5), z1))) MARK(if(div(if(true, div(if(div(y0, y1), y2, y3), y4), y5), z1), z2, z3)) -> c16(MARK(div(if(true, div(if(div(y0, y1), y2, y3), y4), y5), z1))) MARK(if(div(if(true, div(if(if(y0, y1, y2), y3, y4), y5), y6), z1), z2, z3)) -> c16(MARK(div(if(true, div(if(if(y0, y1, y2), y3, y4), y5), y6), z1))) MARK(if(div(if(true, div(if(s(y0), y1, y2), y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, div(if(s(y0), y1, y2), y3), y4), z1))) MARK(if(div(if(true, div(if(true, y0, y1), y2), y3), z1), z2, z3)) -> c16(MARK(div(if(true, div(if(true, y0, y1), y2), y3), z1))) MARK(if(div(if(true, div(if(false, y0, y1), y2), y3), z1), z2, z3)) -> c16(MARK(div(if(true, div(if(false, y0, y1), y2), y3), z1))) MARK(if(div(if(true, div(if(y0, y1, y2), y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, div(if(y0, y1, y2), y3), y4), z1))) MARK(if(div(if(true, if(minus(s(y0), s(y1)), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(minus(s(y0), s(y1)), y2, y3), y4), z1))) MARK(if(div(if(true, if(minus(0, y0), y1, y2), y3), z1), z2, z3)) -> c16(MARK(div(if(true, if(minus(0, y0), y1, y2), y3), z1))) MARK(if(div(if(true, if(minus(y0, y1), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(minus(y0, y1), y2, y3), y4), z1))) MARK(if(div(if(true, if(geq(y0, 0), y1, y2), y3), z1), z2, z3)) -> c16(MARK(div(if(true, if(geq(y0, 0), y1, y2), y3), z1))) MARK(if(div(if(true, if(geq(0, s(y0)), y1, y2), y3), z1), z2, z3)) -> c16(MARK(div(if(true, if(geq(0, s(y0)), y1, y2), y3), z1))) MARK(if(div(if(true, if(geq(s(y0), s(y1)), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(geq(s(y0), s(y1)), y2, y3), y4), z1))) MARK(if(div(if(true, if(geq(y0, y1), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(geq(y0, y1), y2, y3), y4), z1))) MARK(if(div(if(true, if(div(minus(y0, y1), y2), y3, y4), y5), z1), z2, z3)) -> c16(MARK(div(if(true, if(div(minus(y0, y1), y2), y3, y4), y5), z1))) MARK(if(div(if(true, if(div(geq(y0, y1), y2), y3, y4), y5), z1), z2, z3)) -> c16(MARK(div(if(true, if(div(geq(y0, y1), y2), y3, y4), y5), z1))) MARK(if(div(if(true, if(div(div(y0, y1), y2), y3, y4), y5), z1), z2, z3)) -> c16(MARK(div(if(true, if(div(div(y0, y1), y2), y3, y4), y5), z1))) MARK(if(div(if(true, if(div(if(y0, y1, y2), y3), y4, y5), y6), z1), z2, z3)) -> c16(MARK(div(if(true, if(div(if(y0, y1, y2), y3), y4, y5), y6), z1))) MARK(if(div(if(true, if(div(s(y0), y1), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(div(s(y0), y1), y2, y3), y4), z1))) MARK(if(div(if(true, if(div(y0, y1), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(div(y0, y1), y2, y3), y4), z1))) MARK(if(div(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), y6), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), y6), z1))) MARK(if(div(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), y6), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), y6), z1))) MARK(if(div(if(true, if(if(div(y0, y1), y2, y3), y4, y5), y6), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(div(y0, y1), y2, y3), y4, y5), y6), z1))) MARK(if(div(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), y7), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), y7), z1))) MARK(if(div(if(true, if(if(s(y0), y1, y2), y3, y4), y5), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(s(y0), y1, y2), y3, y4), y5), z1))) MARK(if(div(if(true, if(if(true, y0, y1), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(true, y0, y1), y2, y3), y4), z1))) MARK(if(div(if(true, if(if(false, y0, y1), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(false, y0, y1), y2, y3), y4), z1))) MARK(if(div(if(true, if(if(y0, y1, y2), y3, y4), y5), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(y0, y1, y2), y3, y4), y5), z1))) MARK(if(div(if(false, y0, minus(y1, y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, minus(y1, y2)), z1))) MARK(if(div(if(false, y0, geq(y1, y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, geq(y1, y2)), z1))) MARK(if(div(if(false, y0, s(y1)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, s(y1)), z1))) MARK(if(div(if(false, y0, div(s(y1), y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(s(y1), y2)), z1))) MARK(if(div(if(false, y0, if(s(y1), y2, y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(s(y1), y2, y3)), z1))) MARK(if(div(if(false, y0, if(true, y1, y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(true, y1, y2)), z1))) MARK(if(div(if(false, y0, if(false, y1, y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(false, y1, y2)), z1))) MARK(if(div(if(false, y0, div(minus(s(y1), s(y2)), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(minus(s(y1), s(y2)), y3)), z1))) MARK(if(div(if(false, y0, div(minus(0, y1), y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(minus(0, y1), y2)), z1))) MARK(if(div(if(false, y0, div(minus(y1, y2), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(minus(y1, y2), y3)), z1))) MARK(if(div(if(false, y0, div(geq(s(y1), s(y2)), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(geq(s(y1), s(y2)), y3)), z1))) MARK(if(div(if(false, y0, div(geq(y1, 0), y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(geq(y1, 0), y2)), z1))) MARK(if(div(if(false, y0, div(geq(0, s(y1)), y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(geq(0, s(y1)), y2)), z1))) MARK(if(div(if(false, y0, div(geq(y1, y2), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(geq(y1, y2), y3)), z1))) MARK(if(div(if(false, y0, div(div(minus(y1, y2), y3), y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(div(minus(y1, y2), y3), y4)), z1))) MARK(if(div(if(false, y0, div(div(geq(y1, y2), y3), y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(div(geq(y1, y2), y3), y4)), z1))) MARK(if(div(if(false, y0, div(div(div(y1, y2), y3), y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(div(div(y1, y2), y3), y4)), z1))) MARK(if(div(if(false, y0, div(div(if(y1, y2, y3), y4), y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(div(if(y1, y2, y3), y4), y5)), z1))) MARK(if(div(if(false, y0, div(div(s(y1), y2), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(div(s(y1), y2), y3)), z1))) MARK(if(div(if(false, y0, div(div(y1, y2), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(div(y1, y2), y3)), z1))) MARK(if(div(if(false, y0, div(if(minus(y1, y2), y3, y4), y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(minus(y1, y2), y3, y4), y5)), z1))) MARK(if(div(if(false, y0, div(if(geq(y1, y2), y3, y4), y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(geq(y1, y2), y3, y4), y5)), z1))) MARK(if(div(if(false, y0, div(if(div(y1, y2), y3, y4), y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(div(y1, y2), y3, y4), y5)), z1))) MARK(if(div(if(false, y0, div(if(if(y1, y2, y3), y4, y5), y6)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(if(y1, y2, y3), y4, y5), y6)), z1))) MARK(if(div(if(false, y0, div(if(s(y1), y2, y3), y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(s(y1), y2, y3), y4)), z1))) MARK(if(div(if(false, y0, div(if(true, y1, y2), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(true, y1, y2), y3)), z1))) MARK(if(div(if(false, y0, div(if(false, y1, y2), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(false, y1, y2), y3)), z1))) MARK(if(div(if(false, y0, div(if(y1, y2, y3), y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(y1, y2, y3), y4)), z1))) MARK(if(div(if(false, y0, if(minus(s(y1), s(y2)), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(minus(s(y1), s(y2)), y3, y4)), z1))) MARK(if(div(if(false, y0, if(minus(0, y1), y2, y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(minus(0, y1), y2, y3)), z1))) MARK(if(div(if(false, y0, if(minus(y1, y2), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(minus(y1, y2), y3, y4)), z1))) MARK(if(div(if(false, y0, if(geq(y1, 0), y2, y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(geq(y1, 0), y2, y3)), z1))) MARK(if(div(if(false, y0, if(geq(0, s(y1)), y2, y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(geq(0, s(y1)), y2, y3)), z1))) MARK(if(div(if(false, y0, if(geq(s(y1), s(y2)), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(geq(s(y1), s(y2)), y3, y4)), z1))) MARK(if(div(if(false, y0, if(geq(y1, y2), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(geq(y1, y2), y3, y4)), z1))) MARK(if(div(if(false, y0, if(div(minus(y1, y2), y3), y4, y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(div(minus(y1, y2), y3), y4, y5)), z1))) MARK(if(div(if(false, y0, if(div(geq(y1, y2), y3), y4, y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(div(geq(y1, y2), y3), y4, y5)), z1))) MARK(if(div(if(false, y0, if(div(div(y1, y2), y3), y4, y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(div(div(y1, y2), y3), y4, y5)), z1))) MARK(if(div(if(false, y0, if(div(if(y1, y2, y3), y4), y5, y6)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(div(if(y1, y2, y3), y4), y5, y6)), z1))) MARK(if(div(if(false, y0, if(div(s(y1), y2), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(div(s(y1), y2), y3, y4)), z1))) MARK(if(div(if(false, y0, if(div(y1, y2), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(div(y1, y2), y3, y4)), z1))) MARK(if(div(if(false, y0, if(if(minus(y1, y2), y3, y4), y5, y6)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(minus(y1, y2), y3, y4), y5, y6)), z1))) MARK(if(div(if(false, y0, if(if(geq(y1, y2), y3, y4), y5, y6)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(geq(y1, y2), y3, y4), y5, y6)), z1))) MARK(if(div(if(false, y0, if(if(div(y1, y2), y3, y4), y5, y6)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(div(y1, y2), y3, y4), y5, y6)), z1))) MARK(if(div(if(false, y0, if(if(if(y1, y2, y3), y4, y5), y6, y7)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(if(y1, y2, y3), y4, y5), y6, y7)), z1))) MARK(if(div(if(false, y0, if(if(s(y1), y2, y3), y4, y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(s(y1), y2, y3), y4, y5)), z1))) MARK(if(div(if(false, y0, if(if(true, y1, y2), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(true, y1, y2), y3, y4)), z1))) MARK(if(div(if(false, y0, if(if(false, y1, y2), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(false, y1, y2), y3, y4)), z1))) MARK(if(div(if(false, y0, if(if(y1, y2, y3), y4, y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(y1, y2, y3), y4, y5)), z1))) ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: a__minus(0, z0) -> 0 a__minus(s(z0), s(z1)) -> a__minus(z0, z1) a__minus(z0, z1) -> minus(z0, z1) a__geq(z0, 0) -> true a__geq(0, s(z0)) -> false a__geq(s(z0), s(z1)) -> a__geq(z0, z1) a__geq(z0, z1) -> geq(z0, z1) a__div(0, s(z0)) -> 0 a__div(s(z0), s(z1)) -> a__if(a__geq(z0, z1), s(div(minus(z0, z1), s(z1))), 0) a__div(z0, z1) -> div(z0, z1) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) mark(minus(z0, z1)) -> a__minus(z0, z1) mark(geq(z0, z1)) -> a__geq(z0, z1) mark(div(z0, z1)) -> a__div(mark(z0), z1) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) mark(true) -> true mark(false) -> false Tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) MARK(div(div(minus(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__minus(z0, z1), x1), x2), MARK(div(minus(z0, z1), x1))) MARK(div(div(geq(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__geq(z0, z1), x1), x2), MARK(div(geq(z0, z1), x1))) MARK(div(div(div(z0, z1), x1), x2)) -> c15(A__DIV(a__div(a__div(mark(z0), z1), x1), x2), MARK(div(div(z0, z1), x1))) MARK(div(div(if(z0, z1, z2), x1), x2)) -> c15(A__DIV(a__div(a__if(mark(z0), z1, z2), x1), x2), MARK(div(if(z0, z1, z2), x1))) MARK(div(div(s(z0), x1), x2)) -> c15(A__DIV(a__div(s(mark(z0)), x1), x2), MARK(div(s(z0), x1))) MARK(div(if(minus(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__minus(z0, z1), x1, x2), x3), MARK(if(minus(z0, z1), x1, x2))) MARK(div(if(geq(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__geq(z0, z1), x1, x2), x3), MARK(if(geq(z0, z1), x1, x2))) MARK(div(if(div(z0, z1), x1, x2), x3)) -> c15(A__DIV(a__if(a__div(mark(z0), z1), x1, x2), x3), MARK(if(div(z0, z1), x1, x2))) MARK(div(if(if(z0, z1, z2), x1, x2), x3)) -> c15(A__DIV(a__if(a__if(mark(z0), z1, z2), x1, x2), x3), MARK(if(if(z0, z1, z2), x1, x2))) MARK(div(if(s(z0), x1, x2), x3)) -> c15(A__DIV(a__if(s(mark(z0)), x1, x2), x3), MARK(if(s(z0), x1, x2))) MARK(div(if(true, x1, x2), x3)) -> c15(A__DIV(a__if(true, x1, x2), x3), MARK(if(true, x1, x2))) MARK(div(if(false, x1, x2), x3)) -> c15(A__DIV(a__if(false, x1, x2), x3), MARK(if(false, x1, x2))) MARK(if(geq(s(z0), s(z1)), x2, x3)) -> c16(A__IF(a__geq(z0, z1), x2, x3), MARK(geq(s(z0), s(z1)))) MARK(if(div(minus(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__minus(z0, z1), x1), x2, x3), MARK(div(minus(z0, z1), x1))) MARK(if(div(geq(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__geq(z0, z1), x1), x2, x3), MARK(div(geq(z0, z1), x1))) MARK(if(div(div(z0, z1), x1), x2, x3)) -> c16(A__IF(a__div(a__div(mark(z0), z1), x1), x2, x3), MARK(div(div(z0, z1), x1))) MARK(if(div(if(z0, z1, z2), x1), x2, x3)) -> c16(A__IF(a__div(a__if(mark(z0), z1, z2), x1), x2, x3), MARK(div(if(z0, z1, z2), x1))) MARK(if(div(s(z0), x1), x2, x3)) -> c16(A__IF(a__div(s(mark(z0)), x1), x2, x3), MARK(div(s(z0), x1))) MARK(if(if(minus(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__minus(z0, z1), x1, x2), x3, x4), MARK(if(minus(z0, z1), x1, x2))) MARK(if(if(geq(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__geq(z0, z1), x1, x2), x3, x4), MARK(if(geq(z0, z1), x1, x2))) MARK(if(if(div(z0, z1), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__div(mark(z0), z1), x1, x2), x3, x4), MARK(if(div(z0, z1), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c16(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c16(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c16(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c16(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c16(MARK(if(x0, z1, z2))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(geq(z0, 0), x2, x3)) -> c16(A__IF(true, x2, x3)) MARK(if(geq(0, s(z0)), x2, x3)) -> c16(A__IF(false, x2, x3)) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(if(s(s(y0)), z1, z2)) -> c16(MARK(s(s(y0)))) MARK(if(s(div(s(y0), y1)), z1, z2)) -> c16(MARK(s(div(s(y0), y1)))) MARK(if(s(if(s(y0), y1, y2)), z1, z2)) -> c16(MARK(s(if(s(y0), y1, y2)))) MARK(if(s(if(true, y0, y1)), z1, z2)) -> c16(MARK(s(if(true, y0, y1)))) MARK(if(s(if(false, y0, y1)), z1, z2)) -> c16(MARK(s(if(false, y0, y1)))) MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(minus(s(y0), s(y1)), y2)))) MARK(if(s(div(minus(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(minus(y0, y1), y2)))) MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2)) -> c16(MARK(s(div(geq(s(y0), s(y1)), y2)))) MARK(if(s(div(geq(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(geq(y0, y1), y2)))) MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(minus(y0, y1), y2), y3)))) MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(geq(y0, y1), y2), y3)))) MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2)) -> c16(MARK(s(div(div(div(y0, y1), y2), y3)))) MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2)) -> c16(MARK(s(div(div(if(y0, y1, y2), y3), y4)))) MARK(if(s(div(div(s(y0), y1), y2)), z1, z2)) -> c16(MARK(s(div(div(s(y0), y1), y2)))) MARK(if(s(div(div(y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(div(y0, y1), y2)))) MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(minus(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(geq(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2)) -> c16(MARK(s(div(if(div(y0, y1), y2, y3), y4)))) MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2)) -> c16(MARK(s(div(if(if(y0, y1, y2), y3, y4), y5)))) MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(s(y0), y1, y2), y3)))) MARK(if(s(div(if(true, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(true, y0, y1), y2)))) MARK(if(s(div(if(false, y0, y1), y2)), z1, z2)) -> c16(MARK(s(div(if(false, y0, y1), y2)))) MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2)) -> c16(MARK(s(div(if(y0, y1, y2), y3)))) MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(minus(y0, y1), y2, y3)))) MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(s(y0), s(y1)), y2, y3)))) MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(geq(y0, y1), y2, y3)))) MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(minus(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(geq(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(div(div(y0, y1), y2), y3, y4)))) MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(div(if(y0, y1, y2), y3), y4, y5)))) MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(s(y0), y1), y2, y3)))) MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(div(y0, y1), y2, y3)))) MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2)) -> c16(MARK(s(if(if(div(y0, y1), y2, y3), y4, y5)))) MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2)) -> c16(MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)))) MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(s(y0), y1, y2), y3, y4)))) MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(true, y0, y1), y2, y3)))) MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2)) -> c16(MARK(s(if(if(false, y0, y1), y2, y3)))) MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2)) -> c16(MARK(s(if(if(y0, y1, y2), y3, y4)))) MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(minus(s(s(y0)), s(s(y1)))))) MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2)) -> c16(MARK(s(geq(s(s(y0)), s(s(y1)))))) MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(y0, 0), y1, y2)))) MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2)) -> c16(MARK(s(if(geq(0, s(y0)), y1, y2)))) MARK(if(true, minus(y0, y1), z1)) -> c16(A__IF(true, minus(y0, y1), z1)) MARK(if(true, geq(y0, y1), z1)) -> c16(A__IF(true, geq(y0, y1), z1)) MARK(if(true, s(y0), z1)) -> c16(A__IF(true, s(y0), z1)) MARK(if(true, div(s(y0), y1), z1)) -> c16(A__IF(true, div(s(y0), y1), z1)) MARK(if(true, if(s(y0), y1, y2), z1)) -> c16(A__IF(true, if(s(y0), y1, y2), z1)) MARK(if(true, if(true, y0, y1), z1)) -> c16(A__IF(true, if(true, y0, y1), z1)) MARK(if(true, if(false, y0, y1), z1)) -> c16(A__IF(true, if(false, y0, y1), z1)) MARK(if(true, div(minus(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(minus(s(y0), s(y1)), y2), z1)) MARK(if(true, div(minus(0, y0), y1), z1)) -> c16(A__IF(true, div(minus(0, y0), y1), z1)) MARK(if(true, div(minus(y0, y1), y2), z1)) -> c16(A__IF(true, div(minus(y0, y1), y2), z1)) MARK(if(true, div(geq(s(y0), s(y1)), y2), z1)) -> c16(A__IF(true, div(geq(s(y0), s(y1)), y2), z1)) MARK(if(true, div(geq(y0, 0), y1), z1)) -> c16(A__IF(true, div(geq(y0, 0), y1), z1)) MARK(if(true, div(geq(0, s(y0)), y1), z1)) -> c16(A__IF(true, div(geq(0, s(y0)), y1), z1)) MARK(if(true, div(geq(y0, y1), y2), z1)) -> c16(A__IF(true, div(geq(y0, y1), y2), z1)) MARK(if(true, div(div(minus(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(minus(y0, y1), y2), y3), z1)) MARK(if(true, div(div(geq(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(geq(y0, y1), y2), y3), z1)) MARK(if(true, div(div(div(y0, y1), y2), y3), z1)) -> c16(A__IF(true, div(div(div(y0, y1), y2), y3), z1)) MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z1)) -> c16(A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1)) MARK(if(true, div(div(s(y0), y1), y2), z1)) -> c16(A__IF(true, div(div(s(y0), y1), y2), z1)) MARK(if(true, div(div(y0, y1), y2), z1)) -> c16(A__IF(true, div(div(y0, y1), y2), z1)) MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z1)) -> c16(A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1)) MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) -> c16(A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1)) MARK(if(true, div(if(s(y0), y1, y2), y3), z1)) -> c16(A__IF(true, div(if(s(y0), y1, y2), y3), z1)) MARK(if(true, div(if(true, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(true, y0, y1), y2), z1)) MARK(if(true, div(if(false, y0, y1), y2), z1)) -> c16(A__IF(true, div(if(false, y0, y1), y2), z1)) MARK(if(true, div(if(y0, y1, y2), y3), z1)) -> c16(A__IF(true, div(if(y0, y1, y2), y3), z1)) MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(minus(0, y0), y1, y2), z1)) -> c16(A__IF(true, if(minus(0, y0), y1, y2), z1)) MARK(if(true, if(minus(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(minus(y0, y1), y2, y3), z1)) MARK(if(true, if(geq(y0, 0), y1, y2), z1)) -> c16(A__IF(true, if(geq(y0, 0), y1, y2), z1)) MARK(if(true, if(geq(0, s(y0)), y1, y2), z1)) -> c16(A__IF(true, if(geq(0, s(y0)), y1, y2), z1)) MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z1)) -> c16(A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1)) MARK(if(true, if(geq(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(geq(y0, y1), y2, y3), z1)) MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z1)) -> c16(A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1)) MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) -> c16(A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1)) MARK(if(true, if(div(s(y0), y1), y2, y3), z1)) -> c16(A__IF(true, if(div(s(y0), y1), y2, y3), z1)) MARK(if(true, if(div(y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(div(y0, y1), y2, y3), z1)) MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) -> c16(A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1)) MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) -> c16(A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1)) MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1)) MARK(if(true, if(if(true, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(true, y0, y1), y2, y3), z1)) MARK(if(true, if(if(false, y0, y1), y2, y3), z1)) -> c16(A__IF(true, if(if(false, y0, y1), y2, y3), z1)) MARK(if(true, if(if(y0, y1, y2), y3, y4), z1)) -> c16(A__IF(true, if(if(y0, y1, y2), y3, y4), z1)) MARK(if(false, z0, minus(y1, y2))) -> c16(A__IF(false, z0, minus(y1, y2))) MARK(if(false, z0, geq(y1, y2))) -> c16(A__IF(false, z0, geq(y1, y2))) MARK(if(false, z0, s(y1))) -> c16(A__IF(false, z0, s(y1))) MARK(if(false, z0, div(s(y1), y2))) -> c16(A__IF(false, z0, div(s(y1), y2))) MARK(if(false, z0, if(s(y1), y2, y3))) -> c16(A__IF(false, z0, if(s(y1), y2, y3))) MARK(if(false, z0, if(true, y1, y2))) -> c16(A__IF(false, z0, if(true, y1, y2))) MARK(if(false, z0, if(false, y1, y2))) -> c16(A__IF(false, z0, if(false, y1, y2))) MARK(if(false, z0, div(minus(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(minus(s(y1), s(y2)), y3))) MARK(if(false, z0, div(minus(0, y1), y2))) -> c16(A__IF(false, z0, div(minus(0, y1), y2))) MARK(if(false, z0, div(minus(y1, y2), y3))) -> c16(A__IF(false, z0, div(minus(y1, y2), y3))) MARK(if(false, z0, div(geq(s(y1), s(y2)), y3))) -> c16(A__IF(false, z0, div(geq(s(y1), s(y2)), y3))) MARK(if(false, z0, div(geq(y1, 0), y2))) -> c16(A__IF(false, z0, div(geq(y1, 0), y2))) MARK(if(false, z0, div(geq(0, s(y1)), y2))) -> c16(A__IF(false, z0, div(geq(0, s(y1)), y2))) MARK(if(false, z0, div(geq(y1, y2), y3))) -> c16(A__IF(false, z0, div(geq(y1, y2), y3))) MARK(if(false, z0, div(div(minus(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(minus(y1, y2), y3), y4))) MARK(if(false, z0, div(div(geq(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(geq(y1, y2), y3), y4))) MARK(if(false, z0, div(div(div(y1, y2), y3), y4))) -> c16(A__IF(false, z0, div(div(div(y1, y2), y3), y4))) MARK(if(false, z0, div(div(if(y1, y2, y3), y4), y5))) -> c16(A__IF(false, z0, div(div(if(y1, y2, y3), y4), y5))) MARK(if(false, z0, div(div(s(y1), y2), y3))) -> c16(A__IF(false, z0, div(div(s(y1), y2), y3))) MARK(if(false, z0, div(div(y1, y2), y3))) -> c16(A__IF(false, z0, div(div(y1, y2), y3))) MARK(if(false, z0, div(if(minus(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(minus(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(geq(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(geq(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(div(y1, y2), y3, y4), y5))) -> c16(A__IF(false, z0, div(if(div(y1, y2), y3, y4), y5))) MARK(if(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) -> c16(A__IF(false, z0, div(if(if(y1, y2, y3), y4, y5), y6))) MARK(if(false, z0, div(if(s(y1), y2, y3), y4))) -> c16(A__IF(false, z0, div(if(s(y1), y2, y3), y4))) MARK(if(false, z0, div(if(true, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(true, y1, y2), y3))) MARK(if(false, z0, div(if(false, y1, y2), y3))) -> c16(A__IF(false, z0, div(if(false, y1, y2), y3))) MARK(if(false, z0, div(if(y1, y2, y3), y4))) -> c16(A__IF(false, z0, div(if(y1, y2, y3), y4))) MARK(if(false, z0, if(minus(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(minus(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(minus(0, y1), y2, y3))) -> c16(A__IF(false, z0, if(minus(0, y1), y2, y3))) MARK(if(false, z0, if(minus(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(minus(y1, y2), y3, y4))) MARK(if(false, z0, if(geq(y1, 0), y2, y3))) -> c16(A__IF(false, z0, if(geq(y1, 0), y2, y3))) MARK(if(false, z0, if(geq(0, s(y1)), y2, y3))) -> c16(A__IF(false, z0, if(geq(0, s(y1)), y2, y3))) MARK(if(false, z0, if(geq(s(y1), s(y2)), y3, y4))) -> c16(A__IF(false, z0, if(geq(s(y1), s(y2)), y3, y4))) MARK(if(false, z0, if(geq(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(geq(y1, y2), y3, y4))) MARK(if(false, z0, if(div(minus(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(minus(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(geq(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(geq(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(div(y1, y2), y3), y4, y5))) -> c16(A__IF(false, z0, if(div(div(y1, y2), y3), y4, y5))) MARK(if(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) -> c16(A__IF(false, z0, if(div(if(y1, y2, y3), y4), y5, y6))) MARK(if(false, z0, if(div(s(y1), y2), y3, y4))) -> c16(A__IF(false, z0, if(div(s(y1), y2), y3, y4))) MARK(if(false, z0, if(div(y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(div(y1, y2), y3, y4))) MARK(if(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(minus(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(geq(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) -> c16(A__IF(false, z0, if(if(div(y1, y2), y3, y4), y5, y6))) MARK(if(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) -> c16(A__IF(false, z0, if(if(if(y1, y2, y3), y4, y5), y6, y7))) MARK(if(false, z0, if(if(s(y1), y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(s(y1), y2, y3), y4, y5))) MARK(if(false, z0, if(if(true, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(true, y1, y2), y3, y4))) MARK(if(false, z0, if(if(false, y1, y2), y3, y4))) -> c16(A__IF(false, z0, if(if(false, y1, y2), y3, y4))) MARK(if(false, z0, if(if(y1, y2, y3), y4, y5))) -> c16(A__IF(false, z0, if(if(y1, y2, y3), y4, y5))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__geq(s(y0), s(y1)), z2), MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(geq(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(geq(s(s(y0)), s(s(y1))))) MARK(div(div(s(y0), z1), z2)) -> c15(MARK(div(s(y0), z1))) MARK(div(div(div(minus(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(minus(y0, y1), y2), z1))) MARK(div(div(div(geq(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(geq(y0, y1), y2), z1))) MARK(div(div(div(div(y0, y1), y2), z1), z2)) -> c15(MARK(div(div(div(y0, y1), y2), z1))) MARK(div(div(div(if(y0, y1, y2), y3), z1), z2)) -> c15(MARK(div(div(if(y0, y1, y2), y3), z1))) MARK(div(div(div(s(y0), y1), z1), z2)) -> c15(MARK(div(div(s(y0), y1), z1))) MARK(div(div(div(y0, y1), z1), z2)) -> c15(MARK(div(div(y0, y1), z1))) MARK(div(div(if(minus(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(minus(y0, y1), y2, y3), z1))) MARK(div(div(if(geq(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(geq(y0, y1), y2, y3), z1))) MARK(div(div(if(div(y0, y1), y2, y3), z1), z2)) -> c15(MARK(div(if(div(y0, y1), y2, y3), z1))) MARK(div(div(if(if(y0, y1, y2), y3, y4), z1), z2)) -> c15(MARK(div(if(if(y0, y1, y2), y3, y4), z1))) MARK(div(div(if(s(y0), y1, y2), z1), z2)) -> c15(MARK(div(if(s(y0), y1, y2), z1))) MARK(div(div(if(true, y0, y1), z1), z2)) -> c15(MARK(div(if(true, y0, y1), z1))) MARK(div(div(if(false, y0, y1), z1), z2)) -> c15(MARK(div(if(false, y0, y1), z1))) MARK(div(div(if(y0, y1, y2), z1), z2)) -> c15(MARK(div(if(y0, y1, y2), z1))) MARK(div(div(minus(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(minus(s(s(y0)), s(s(y1))), z1))) MARK(div(div(geq(s(s(y0)), s(s(y1))), z1), z2)) -> c15(MARK(div(geq(s(s(y0)), s(s(y1))), z1))) MARK(div(if(minus(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(minus(s(y0), s(y1)), z1, z2))) MARK(div(if(minus(y0, y1), z1, z2), z3)) -> c15(MARK(if(minus(y0, y1), z1, z2))) MARK(div(if(geq(s(y0), s(y1)), z1, z2), z3)) -> c15(MARK(if(geq(s(y0), s(y1)), z1, z2))) MARK(div(if(geq(y0, y1), z1, z2), z3)) -> c15(MARK(if(geq(y0, y1), z1, z2))) MARK(div(if(div(minus(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(minus(y0, y1), y2), z1, z2))) MARK(div(if(div(geq(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(geq(y0, y1), y2), z1, z2))) MARK(div(if(div(div(y0, y1), y2), z1, z2), z3)) -> c15(MARK(if(div(div(y0, y1), y2), z1, z2))) MARK(div(if(div(if(y0, y1, y2), y3), z1, z2), z3)) -> c15(MARK(if(div(if(y0, y1, y2), y3), z1, z2))) MARK(div(if(div(s(y0), y1), z1, z2), z3)) -> c15(MARK(if(div(s(y0), y1), z1, z2))) MARK(div(if(div(y0, y1), z1, z2), z3)) -> c15(MARK(if(div(y0, y1), z1, z2))) MARK(div(if(if(minus(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(minus(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(geq(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(geq(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(div(y0, y1), y2, y3), z1, z2), z3)) -> c15(MARK(if(if(div(y0, y1), y2, y3), z1, z2))) MARK(div(if(if(if(y0, y1, y2), y3, y4), z1, z2), z3)) -> c15(MARK(if(if(if(y0, y1, y2), y3, y4), z1, z2))) MARK(div(if(if(s(y0), y1, y2), z1, z2), z3)) -> c15(MARK(if(if(s(y0), y1, y2), z1, z2))) MARK(div(if(if(true, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(true, y0, y1), z1, z2))) MARK(div(if(if(false, y0, y1), z1, z2), z3)) -> c15(MARK(if(if(false, y0, y1), z1, z2))) MARK(div(if(if(y0, y1, y2), z1, z2), z3)) -> c15(MARK(if(if(y0, y1, y2), z1, z2))) MARK(div(if(geq(y0, 0), z1, z2), z3)) -> c15(MARK(if(geq(y0, 0), z1, z2))) MARK(div(if(geq(0, s(y0)), z1, z2), z3)) -> c15(MARK(if(geq(0, s(y0)), z1, z2))) MARK(div(if(s(s(y0)), z1, z2), z3)) -> c15(MARK(if(s(s(y0)), z1, z2))) MARK(div(if(s(div(s(y0), y1)), z1, z2), z3)) -> c15(MARK(if(s(div(s(y0), y1)), z1, z2))) MARK(div(if(s(if(s(y0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(s(y0), y1, y2)), z1, z2))) MARK(div(if(s(if(true, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(true, y0, y1)), z1, z2))) MARK(div(if(s(if(false, y0, y1)), z1, z2), z3)) -> c15(MARK(if(s(if(false, y0, y1)), z1, z2))) MARK(div(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(minus(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(minus(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(s(y0), s(y1)), y2)), z1, z2))) MARK(div(if(s(div(geq(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(geq(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(minus(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(geq(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(div(y0, y1), y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(div(div(y0, y1), y2), y3)), z1, z2))) MARK(div(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(div(if(y0, y1, y2), y3), y4)), z1, z2))) MARK(div(if(s(div(div(s(y0), y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(s(y0), y1), y2)), z1, z2))) MARK(div(if(s(div(div(y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(div(y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(minus(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(geq(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2), z3)) -> c15(MARK(if(s(div(if(div(y0, y1), y2, y3), y4)), z1, z2))) MARK(div(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2), z3)) -> c15(MARK(if(s(div(if(if(y0, y1, y2), y3, y4), y5)), z1, z2))) MARK(div(if(s(div(if(s(y0), y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(s(y0), y1, y2), y3)), z1, z2))) MARK(div(if(s(div(if(true, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(true, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(false, y0, y1), y2)), z1, z2), z3)) -> c15(MARK(if(s(div(if(false, y0, y1), y2)), z1, z2))) MARK(div(if(s(div(if(y0, y1, y2), y3)), z1, z2), z3)) -> c15(MARK(if(s(div(if(y0, y1, y2), y3)), z1, z2))) MARK(div(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(minus(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(minus(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(s(y0), s(y1)), y2, y3)), z1, z2))) MARK(div(if(s(if(geq(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(minus(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(geq(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(div(div(y0, y1), y2), y3, y4)), z1, z2))) MARK(div(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(div(if(y0, y1, y2), y3), y4, y5)), z1, z2))) MARK(div(if(s(if(div(s(y0), y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(s(y0), y1), y2, y3)), z1, z2))) MARK(div(if(s(if(div(y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(div(y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(minus(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(geq(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2), z3)) -> c15(MARK(if(s(if(if(div(y0, y1), y2, y3), y4, y5)), z1, z2))) MARK(div(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2), z3)) -> c15(MARK(if(s(if(if(if(y0, y1, y2), y3, y4), y5, y6)), z1, z2))) MARK(div(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(s(y0), y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(if(if(true, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(true, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(false, y0, y1), y2, y3)), z1, z2), z3)) -> c15(MARK(if(s(if(if(false, y0, y1), y2, y3)), z1, z2))) MARK(div(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2), z3)) -> c15(MARK(if(s(if(if(y0, y1, y2), y3, y4)), z1, z2))) MARK(div(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(minus(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2), z3)) -> c15(MARK(if(s(geq(s(s(y0)), s(s(y1)))), z1, z2))) MARK(div(if(s(if(geq(y0, 0), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(y0, 0), y1, y2)), z1, z2))) MARK(div(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2), z3)) -> c15(MARK(if(s(if(geq(0, s(y0)), y1, y2)), z1, z2))) MARK(div(if(true, minus(y0, y1), z2), z3)) -> c15(MARK(if(true, minus(y0, y1), z2))) MARK(div(if(true, geq(y0, y1), z2), z3)) -> c15(MARK(if(true, geq(y0, y1), z2))) MARK(div(if(true, s(y0), z2), z3)) -> c15(MARK(if(true, s(y0), z2))) MARK(div(if(true, div(s(y0), y1), z2), z3)) -> c15(MARK(if(true, div(s(y0), y1), z2))) MARK(div(if(true, if(s(y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(s(y0), y1, y2), z2))) MARK(div(if(true, if(true, y0, y1), z2), z3)) -> c15(MARK(if(true, if(true, y0, y1), z2))) MARK(div(if(true, if(false, y0, y1), z2), z3)) -> c15(MARK(if(true, if(false, y0, y1), z2))) MARK(div(if(true, div(minus(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(minus(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(minus(0, y0), y1), z2), z3)) -> c15(MARK(if(true, div(minus(0, y0), y1), z2))) MARK(div(if(true, div(minus(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(minus(y0, y1), y2), z2))) MARK(div(if(true, div(geq(s(y0), s(y1)), y2), z2), z3)) -> c15(MARK(if(true, div(geq(s(y0), s(y1)), y2), z2))) MARK(div(if(true, div(geq(y0, 0), y1), z2), z3)) -> c15(MARK(if(true, div(geq(y0, 0), y1), z2))) MARK(div(if(true, div(geq(0, s(y0)), y1), z2), z3)) -> c15(MARK(if(true, div(geq(0, s(y0)), y1), z2))) MARK(div(if(true, div(geq(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(geq(y0, y1), y2), z2))) MARK(div(if(true, div(div(minus(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(minus(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(geq(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(geq(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(div(y0, y1), y2), y3), z2), z3)) -> c15(MARK(if(true, div(div(div(y0, y1), y2), y3), z2))) MARK(div(if(true, div(div(if(y0, y1, y2), y3), y4), z2), z3)) -> c15(MARK(if(true, div(div(if(y0, y1, y2), y3), y4), z2))) MARK(div(if(true, div(div(s(y0), y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(s(y0), y1), y2), z2))) MARK(div(if(true, div(div(y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(div(y0, y1), y2), z2))) MARK(div(if(true, div(if(minus(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(minus(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(geq(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(geq(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(div(y0, y1), y2, y3), y4), z2), z3)) -> c15(MARK(if(true, div(if(div(y0, y1), y2, y3), y4), z2))) MARK(div(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2), z3)) -> c15(MARK(if(true, div(if(if(y0, y1, y2), y3, y4), y5), z2))) MARK(div(if(true, div(if(s(y0), y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(s(y0), y1, y2), y3), z2))) MARK(div(if(true, div(if(true, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(true, y0, y1), y2), z2))) MARK(div(if(true, div(if(false, y0, y1), y2), z2), z3)) -> c15(MARK(if(true, div(if(false, y0, y1), y2), z2))) MARK(div(if(true, div(if(y0, y1, y2), y3), z2), z3)) -> c15(MARK(if(true, div(if(y0, y1, y2), y3), z2))) MARK(div(if(true, if(minus(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(minus(0, y0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(minus(0, y0), y1, y2), z2))) MARK(div(if(true, if(minus(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(minus(y0, y1), y2, y3), z2))) MARK(div(if(true, if(geq(y0, 0), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(y0, 0), y1, y2), z2))) MARK(div(if(true, if(geq(0, s(y0)), y1, y2), z2), z3)) -> c15(MARK(if(true, if(geq(0, s(y0)), y1, y2), z2))) MARK(div(if(true, if(geq(s(y0), s(y1)), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(s(y0), s(y1)), y2, y3), z2))) MARK(div(if(true, if(geq(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(geq(y0, y1), y2, y3), z2))) MARK(div(if(true, if(div(minus(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(minus(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(geq(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(geq(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(div(y0, y1), y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(div(div(y0, y1), y2), y3, y4), z2))) MARK(div(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(div(if(y0, y1, y2), y3), y4, y5), z2))) MARK(div(if(true, if(div(s(y0), y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(s(y0), y1), y2, y3), z2))) MARK(div(if(true, if(div(y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(div(y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2), z3)) -> c15(MARK(if(true, if(if(div(y0, y1), y2, y3), y4, y5), z2))) MARK(div(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2), z3)) -> c15(MARK(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z2))) MARK(div(if(true, if(if(s(y0), y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(s(y0), y1, y2), y3, y4), z2))) MARK(div(if(true, if(if(true, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(true, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(false, y0, y1), y2, y3), z2), z3)) -> c15(MARK(if(true, if(if(false, y0, y1), y2, y3), z2))) MARK(div(if(true, if(if(y0, y1, y2), y3, y4), z2), z3)) -> c15(MARK(if(true, if(if(y0, y1, y2), y3, y4), z2))) MARK(div(if(false, z1, minus(y1, y2)), z3)) -> c15(MARK(if(false, z1, minus(y1, y2)))) MARK(div(if(false, z1, geq(y1, y2)), z3)) -> c15(MARK(if(false, z1, geq(y1, y2)))) MARK(div(if(false, z1, s(y1)), z3)) -> c15(MARK(if(false, z1, s(y1)))) MARK(div(if(false, z1, div(s(y1), y2)), z3)) -> c15(MARK(if(false, z1, div(s(y1), y2)))) MARK(div(if(false, z1, if(s(y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(s(y1), y2, y3)))) MARK(div(if(false, z1, if(true, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(true, y1, y2)))) MARK(div(if(false, z1, if(false, y1, y2)), z3)) -> c15(MARK(if(false, z1, if(false, y1, y2)))) MARK(div(if(false, z1, div(minus(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(minus(0, y1), y2)), z3)) -> c15(MARK(if(false, z1, div(minus(0, y1), y2)))) MARK(div(if(false, z1, div(minus(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(minus(y1, y2), y3)))) MARK(div(if(false, z1, div(geq(s(y1), s(y2)), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(s(y1), s(y2)), y3)))) MARK(div(if(false, z1, div(geq(y1, 0), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, 0), y2)))) MARK(div(if(false, z1, div(geq(0, s(y1)), y2)), z3)) -> c15(MARK(if(false, z1, div(geq(0, s(y1)), y2)))) MARK(div(if(false, z1, div(geq(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(geq(y1, y2), y3)))) MARK(div(if(false, z1, div(div(minus(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(minus(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(geq(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(geq(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(div(y1, y2), y3), y4)), z3)) -> c15(MARK(if(false, z1, div(div(div(y1, y2), y3), y4)))) MARK(div(if(false, z1, div(div(if(y1, y2, y3), y4), y5)), z3)) -> c15(MARK(if(false, z1, div(div(if(y1, y2, y3), y4), y5)))) MARK(div(if(false, z1, div(div(s(y1), y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(s(y1), y2), y3)))) MARK(div(if(false, z1, div(div(y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(div(y1, y2), y3)))) MARK(div(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(minus(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(geq(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(div(y1, y2), y3, y4), y5)), z3)) -> c15(MARK(if(false, z1, div(if(div(y1, y2), y3, y4), y5)))) MARK(div(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)), z3)) -> c15(MARK(if(false, z1, div(if(if(y1, y2, y3), y4, y5), y6)))) MARK(div(if(false, z1, div(if(s(y1), y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(s(y1), y2, y3), y4)))) MARK(div(if(false, z1, div(if(true, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(true, y1, y2), y3)))) MARK(div(if(false, z1, div(if(false, y1, y2), y3)), z3)) -> c15(MARK(if(false, z1, div(if(false, y1, y2), y3)))) MARK(div(if(false, z1, div(if(y1, y2, y3), y4)), z3)) -> c15(MARK(if(false, z1, div(if(y1, y2, y3), y4)))) MARK(div(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(minus(0, y1), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(minus(0, y1), y2, y3)))) MARK(div(if(false, z1, if(minus(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(minus(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, 0), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, 0), y2, y3)))) MARK(div(if(false, z1, if(geq(0, s(y1)), y2, y3)), z3)) -> c15(MARK(if(false, z1, if(geq(0, s(y1)), y2, y3)))) MARK(div(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(s(y1), s(y2)), y3, y4)))) MARK(div(if(false, z1, if(geq(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(geq(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(minus(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(geq(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(div(y1, y2), y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(div(div(y1, y2), y3), y4, y5)))) MARK(div(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(div(if(y1, y2, y3), y4), y5, y6)))) MARK(div(if(false, z1, if(div(s(y1), y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(s(y1), y2), y3, y4)))) MARK(div(if(false, z1, if(div(y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(div(y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(minus(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(geq(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)), z3)) -> c15(MARK(if(false, z1, if(if(div(y1, y2), y3, y4), y5, y6)))) MARK(div(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)), z3)) -> c15(MARK(if(false, z1, if(if(if(y1, y2, y3), y4, y5), y6, y7)))) MARK(div(if(false, z1, if(if(s(y1), y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(s(y1), y2, y3), y4, y5)))) MARK(div(if(false, z1, if(if(true, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(true, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(false, y1, y2), y3, y4)), z3)) -> c15(MARK(if(false, z1, if(if(false, y1, y2), y3, y4)))) MARK(div(if(false, z1, if(if(y1, y2, y3), y4, y5)), z3)) -> c15(MARK(if(false, z1, if(if(y1, y2, y3), y4, y5)))) MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(A__IF(a__minus(s(y0), s(y1)), z2, z3), MARK(minus(s(s(y0)), s(s(y1))))) MARK(if(minus(s(s(y0)), s(s(y1))), z2, z3)) -> c16(MARK(minus(s(s(y0)), s(s(y1))))) MARK(if(geq(s(s(y0)), s(s(y1))), z2, z3)) -> c16(MARK(geq(s(s(y0)), s(s(y1))))) MARK(if(div(s(y0), z1), z2, z3)) -> c16(MARK(div(s(y0), z1))) MARK(if(div(div(minus(y0, y1), y2), z1), z2, z3)) -> c16(MARK(div(div(minus(y0, y1), y2), z1))) MARK(if(div(div(geq(y0, y1), y2), z1), z2, z3)) -> c16(MARK(div(div(geq(y0, y1), y2), z1))) MARK(if(div(div(div(y0, y1), y2), z1), z2, z3)) -> c16(MARK(div(div(div(y0, y1), y2), z1))) MARK(if(div(div(if(y0, y1, y2), y3), z1), z2, z3)) -> c16(MARK(div(div(if(y0, y1, y2), y3), z1))) MARK(if(div(div(s(y0), y1), z1), z2, z3)) -> c16(MARK(div(div(s(y0), y1), z1))) MARK(if(div(if(minus(y0, y1), y2, y3), z1), z2, z3)) -> c16(MARK(div(if(minus(y0, y1), y2, y3), z1))) MARK(if(div(if(geq(y0, y1), y2, y3), z1), z2, z3)) -> c16(MARK(div(if(geq(y0, y1), y2, y3), z1))) MARK(if(div(if(div(y0, y1), y2, y3), z1), z2, z3)) -> c16(MARK(div(if(div(y0, y1), y2, y3), z1))) MARK(if(div(if(if(y0, y1, y2), y3, y4), z1), z2, z3)) -> c16(MARK(div(if(if(y0, y1, y2), y3, y4), z1))) MARK(if(div(if(s(y0), y1, y2), z1), z2, z3)) -> c16(MARK(div(if(s(y0), y1, y2), z1))) MARK(if(div(if(true, y0, y1), z1), z2, z3)) -> c16(MARK(div(if(true, y0, y1), z1))) MARK(if(div(if(false, y0, y1), z1), z2, z3)) -> c16(MARK(div(if(false, y0, y1), z1))) MARK(if(div(minus(s(s(y0)), s(s(y1))), z1), z2, z3)) -> c16(MARK(div(minus(s(s(y0)), s(s(y1))), z1))) MARK(if(div(geq(s(s(y0)), s(s(y1))), z1), z2, z3)) -> c16(MARK(div(geq(s(s(y0)), s(s(y1))), z1))) MARK(if(div(div(div(minus(y0, y1), y2), y3), z1), z2, z3)) -> c16(MARK(div(div(div(minus(y0, y1), y2), y3), z1))) MARK(if(div(div(div(geq(y0, 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y5), z1))) MARK(if(div(if(true, div(if(geq(y0, y1), y2, y3), y4), y5), z1), z2, z3)) -> c16(MARK(div(if(true, div(if(geq(y0, y1), y2, y3), y4), y5), z1))) MARK(if(div(if(true, div(if(div(y0, y1), y2, y3), y4), y5), z1), z2, z3)) -> c16(MARK(div(if(true, div(if(div(y0, y1), y2, y3), y4), y5), z1))) MARK(if(div(if(true, div(if(if(y0, y1, y2), y3, y4), y5), y6), z1), z2, z3)) -> c16(MARK(div(if(true, div(if(if(y0, y1, y2), y3, y4), y5), y6), z1))) MARK(if(div(if(true, div(if(s(y0), y1, y2), y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, div(if(s(y0), y1, y2), y3), y4), z1))) MARK(if(div(if(true, div(if(true, y0, y1), y2), y3), z1), z2, z3)) -> c16(MARK(div(if(true, div(if(true, y0, y1), y2), y3), z1))) MARK(if(div(if(true, div(if(false, y0, y1), y2), y3), z1), z2, z3)) -> c16(MARK(div(if(true, div(if(false, y0, y1), y2), y3), z1))) MARK(if(div(if(true, div(if(y0, y1, y2), y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, div(if(y0, y1, y2), y3), y4), z1))) MARK(if(div(if(true, if(minus(s(y0), s(y1)), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(minus(s(y0), s(y1)), y2, y3), y4), z1))) MARK(if(div(if(true, if(minus(0, y0), y1, y2), y3), z1), z2, z3)) -> c16(MARK(div(if(true, if(minus(0, y0), y1, y2), y3), z1))) MARK(if(div(if(true, if(minus(y0, y1), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(minus(y0, y1), y2, y3), y4), z1))) MARK(if(div(if(true, if(geq(y0, 0), y1, y2), y3), z1), z2, z3)) -> c16(MARK(div(if(true, if(geq(y0, 0), y1, y2), y3), z1))) MARK(if(div(if(true, if(geq(0, s(y0)), y1, y2), y3), z1), z2, z3)) -> c16(MARK(div(if(true, if(geq(0, s(y0)), y1, y2), y3), z1))) MARK(if(div(if(true, if(geq(s(y0), s(y1)), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(geq(s(y0), s(y1)), y2, y3), y4), z1))) MARK(if(div(if(true, if(geq(y0, y1), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(geq(y0, y1), y2, y3), y4), z1))) MARK(if(div(if(true, if(div(minus(y0, y1), y2), y3, y4), y5), z1), z2, z3)) -> c16(MARK(div(if(true, if(div(minus(y0, y1), y2), y3, y4), y5), z1))) MARK(if(div(if(true, if(div(geq(y0, y1), y2), y3, y4), y5), z1), z2, z3)) -> c16(MARK(div(if(true, if(div(geq(y0, y1), y2), y3, y4), y5), z1))) MARK(if(div(if(true, if(div(div(y0, y1), y2), y3, y4), y5), z1), z2, z3)) -> c16(MARK(div(if(true, if(div(div(y0, y1), y2), y3, y4), y5), z1))) MARK(if(div(if(true, if(div(if(y0, y1, y2), y3), y4, y5), y6), z1), z2, z3)) -> c16(MARK(div(if(true, if(div(if(y0, y1, y2), y3), y4, y5), y6), z1))) MARK(if(div(if(true, if(div(s(y0), y1), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(div(s(y0), y1), y2, y3), y4), z1))) MARK(if(div(if(true, if(div(y0, y1), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(div(y0, y1), y2, y3), y4), z1))) MARK(if(div(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), y6), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(minus(y0, y1), y2, y3), y4, y5), y6), z1))) MARK(if(div(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), y6), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(geq(y0, y1), y2, y3), y4, y5), y6), z1))) MARK(if(div(if(true, if(if(div(y0, y1), y2, y3), y4, y5), y6), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(div(y0, y1), y2, y3), y4, y5), y6), z1))) MARK(if(div(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), y7), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), y7), z1))) MARK(if(div(if(true, if(if(s(y0), y1, y2), y3, y4), y5), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(s(y0), y1, y2), y3, y4), y5), z1))) MARK(if(div(if(true, if(if(true, y0, y1), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(true, y0, y1), y2, y3), y4), z1))) MARK(if(div(if(true, if(if(false, y0, y1), y2, y3), y4), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(false, y0, y1), y2, y3), y4), z1))) MARK(if(div(if(true, if(if(y0, y1, y2), y3, y4), y5), z1), z2, z3)) -> c16(MARK(div(if(true, if(if(y0, y1, y2), y3, y4), y5), z1))) MARK(if(div(if(false, y0, minus(y1, y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, minus(y1, y2)), z1))) MARK(if(div(if(false, y0, geq(y1, y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, geq(y1, y2)), z1))) MARK(if(div(if(false, y0, s(y1)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, s(y1)), z1))) MARK(if(div(if(false, y0, div(s(y1), y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(s(y1), y2)), z1))) MARK(if(div(if(false, y0, if(s(y1), y2, y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(s(y1), y2, y3)), z1))) MARK(if(div(if(false, y0, if(true, y1, y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(true, y1, y2)), z1))) MARK(if(div(if(false, y0, if(false, y1, y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(false, y1, y2)), z1))) MARK(if(div(if(false, y0, div(minus(s(y1), s(y2)), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(minus(s(y1), s(y2)), y3)), z1))) MARK(if(div(if(false, y0, div(minus(0, y1), y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(minus(0, y1), y2)), z1))) MARK(if(div(if(false, y0, div(minus(y1, y2), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(minus(y1, y2), y3)), z1))) MARK(if(div(if(false, y0, div(geq(s(y1), s(y2)), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(geq(s(y1), s(y2)), y3)), z1))) MARK(if(div(if(false, y0, div(geq(y1, 0), y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(geq(y1, 0), y2)), z1))) MARK(if(div(if(false, y0, div(geq(0, s(y1)), y2)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(geq(0, s(y1)), y2)), z1))) MARK(if(div(if(false, y0, div(geq(y1, y2), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(geq(y1, y2), y3)), z1))) MARK(if(div(if(false, y0, div(div(minus(y1, y2), y3), y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(div(minus(y1, y2), y3), y4)), z1))) MARK(if(div(if(false, y0, div(div(geq(y1, y2), y3), y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(div(geq(y1, y2), y3), y4)), z1))) MARK(if(div(if(false, y0, div(div(div(y1, y2), y3), y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(div(div(y1, y2), y3), y4)), z1))) MARK(if(div(if(false, y0, div(div(if(y1, y2, y3), y4), y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(div(if(y1, y2, y3), y4), y5)), z1))) MARK(if(div(if(false, y0, div(div(s(y1), y2), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(div(s(y1), y2), y3)), z1))) MARK(if(div(if(false, y0, div(div(y1, y2), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(div(y1, y2), y3)), z1))) MARK(if(div(if(false, y0, div(if(minus(y1, y2), y3, y4), y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(minus(y1, y2), y3, y4), y5)), z1))) MARK(if(div(if(false, y0, div(if(geq(y1, y2), y3, y4), y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(geq(y1, y2), y3, y4), y5)), z1))) MARK(if(div(if(false, y0, div(if(div(y1, y2), y3, y4), y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(div(y1, y2), y3, y4), y5)), z1))) MARK(if(div(if(false, y0, div(if(if(y1, y2, y3), y4, y5), y6)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(if(y1, y2, y3), y4, y5), y6)), z1))) MARK(if(div(if(false, y0, div(if(s(y1), y2, y3), y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(s(y1), y2, y3), y4)), z1))) MARK(if(div(if(false, y0, div(if(true, y1, y2), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(true, y1, y2), y3)), z1))) MARK(if(div(if(false, y0, div(if(false, y1, y2), y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(false, y1, y2), y3)), z1))) MARK(if(div(if(false, y0, div(if(y1, y2, y3), y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, div(if(y1, y2, y3), y4)), z1))) MARK(if(div(if(false, y0, if(minus(s(y1), s(y2)), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(minus(s(y1), s(y2)), y3, y4)), z1))) MARK(if(div(if(false, y0, if(minus(0, y1), y2, y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(minus(0, y1), y2, y3)), z1))) MARK(if(div(if(false, y0, if(minus(y1, y2), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(minus(y1, y2), y3, y4)), z1))) MARK(if(div(if(false, y0, if(geq(y1, 0), y2, y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(geq(y1, 0), y2, y3)), z1))) MARK(if(div(if(false, y0, if(geq(0, s(y1)), y2, y3)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(geq(0, s(y1)), y2, y3)), z1))) MARK(if(div(if(false, y0, if(geq(s(y1), s(y2)), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(geq(s(y1), s(y2)), y3, y4)), z1))) MARK(if(div(if(false, y0, if(geq(y1, y2), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(geq(y1, y2), y3, y4)), z1))) MARK(if(div(if(false, y0, if(div(minus(y1, y2), y3), y4, y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(div(minus(y1, y2), y3), y4, y5)), z1))) MARK(if(div(if(false, y0, if(div(geq(y1, y2), y3), y4, y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(div(geq(y1, y2), y3), y4, y5)), z1))) MARK(if(div(if(false, y0, if(div(div(y1, y2), y3), y4, y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(div(div(y1, y2), y3), y4, y5)), z1))) MARK(if(div(if(false, y0, if(div(if(y1, y2, y3), y4), y5, y6)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(div(if(y1, y2, y3), y4), y5, y6)), z1))) MARK(if(div(if(false, y0, if(div(s(y1), y2), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(div(s(y1), y2), y3, y4)), z1))) MARK(if(div(if(false, y0, if(div(y1, y2), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(div(y1, y2), y3, y4)), z1))) MARK(if(div(if(false, y0, if(if(minus(y1, y2), y3, y4), y5, y6)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(minus(y1, y2), y3, y4), y5, y6)), z1))) MARK(if(div(if(false, y0, if(if(geq(y1, y2), y3, y4), y5, y6)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(geq(y1, y2), y3, y4), y5, y6)), z1))) MARK(if(div(if(false, y0, if(if(div(y1, y2), y3, y4), y5, y6)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(div(y1, y2), y3, y4), y5, y6)), z1))) MARK(if(div(if(false, y0, if(if(if(y1, y2, y3), y4, y5), y6, y7)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(if(y1, y2, y3), y4, y5), y6, y7)), z1))) MARK(if(div(if(false, y0, if(if(s(y1), y2, y3), y4, y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(s(y1), y2, y3), y4, y5)), z1))) MARK(if(div(if(false, y0, if(if(true, y1, y2), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(true, y1, y2), y3, y4)), z1))) MARK(if(div(if(false, y0, if(if(false, y1, y2), y3, y4)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(false, y1, y2), y3, y4)), z1))) MARK(if(div(if(false, y0, if(if(y1, y2, y3), y4, y5)), z1), z2, z3)) -> c16(MARK(div(if(false, y0, if(if(y1, y2, y3), y4, y5)), z1))) S tuples: A__DIV(s(z0), s(0)) -> c8(A__IF(true, s(div(minus(z0, 0), s(0))), 0)) MARK(div(s(z0), x1)) -> c15(A__DIV(s(mark(z0)), x1), MARK(s(z0))) A__DIV(s(s(s(z0))), s(s(s(z1)))) -> c8(A__IF(a__geq(z0, z1), s(div(minus(s(s(z0)), s(s(z1))), s(s(s(z1))))), 0), A__GEQ(s(s(z0)), s(s(z1)))) A__MINUS(s(s(y0)), s(s(y1))) -> c1(A__MINUS(s(y0), s(y1))) A__GEQ(s(s(y0)), s(s(y1))) -> c5(A__GEQ(s(y0), s(y1))) A__DIV(s(s(z0)), s(s(0))) -> c8(A__IF(true, s(div(minus(s(z0), s(0)), s(s(0)))), 0)) A__IF(true, minus(y0, y1), z1) -> c10(MARK(minus(y0, y1))) A__IF(true, geq(y0, y1), z1) -> c10(MARK(geq(y0, y1))) A__IF(true, s(y0), z1) -> c10(MARK(s(y0))) A__IF(true, div(s(y0), y1), z1) -> c10(MARK(div(s(y0), y1))) A__IF(true, if(s(y0), y1, y2), z1) -> c10(MARK(if(s(y0), y1, y2))) A__IF(true, if(true, y0, y1), z1) -> c10(MARK(if(true, y0, y1))) A__IF(true, if(false, y0, y1), z1) -> c10(MARK(if(false, y0, y1))) A__IF(true, div(minus(s(y0), s(y1)), y2), z1) -> c10(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(true, div(minus(y0, y1), y2), z1) -> c10(MARK(div(minus(y0, y1), y2))) A__IF(true, div(geq(s(y0), s(y1)), y2), z1) -> c10(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(true, div(geq(y0, y1), y2), z1) -> c10(MARK(div(geq(y0, y1), y2))) A__IF(true, div(div(minus(y0, y1), y2), y3), z1) -> c10(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(true, div(div(geq(y0, y1), y2), y3), z1) -> c10(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(true, div(div(div(y0, y1), y2), y3), z1) -> c10(MARK(div(div(div(y0, y1), y2), y3))) A__IF(true, div(div(if(y0, y1, y2), y3), y4), z1) -> c10(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(true, div(div(s(y0), y1), y2), z1) -> c10(MARK(div(div(s(y0), y1), y2))) A__IF(true, div(div(y0, y1), y2), z1) -> c10(MARK(div(div(y0, y1), y2))) A__IF(true, div(if(minus(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(true, div(if(geq(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(true, div(if(div(y0, y1), y2, y3), y4), z1) -> c10(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(true, div(if(if(y0, y1, y2), y3, y4), y5), z1) -> c10(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(true, div(if(s(y0), y1, y2), y3), z1) -> c10(MARK(div(if(s(y0), y1, y2), y3))) A__IF(true, div(if(true, y0, y1), y2), z1) -> c10(MARK(div(if(true, y0, y1), y2))) A__IF(true, div(if(false, y0, y1), y2), z1) -> c10(MARK(div(if(false, y0, y1), y2))) A__IF(true, div(if(y0, y1, y2), y3), z1) -> c10(MARK(div(if(y0, y1, y2), y3))) A__IF(true, if(minus(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(true, if(minus(y0, y1), y2, y3), z1) -> c10(MARK(if(minus(y0, y1), y2, y3))) A__IF(true, if(geq(y0, 0), y1, y2), z1) -> c10(MARK(if(geq(y0, 0), y1, y2))) A__IF(true, if(geq(0, s(y0)), y1, y2), z1) -> c10(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(true, if(geq(s(y0), s(y1)), y2, y3), z1) -> c10(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(true, if(geq(y0, y1), y2, y3), z1) -> c10(MARK(if(geq(y0, y1), y2, y3))) A__IF(true, if(div(minus(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(true, if(div(geq(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(true, if(div(div(y0, y1), y2), y3, y4), z1) -> c10(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(true, if(div(if(y0, y1, y2), y3), y4, y5), z1) -> c10(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(true, if(div(s(y0), y1), y2, y3), z1) -> c10(MARK(if(div(s(y0), y1), y2, y3))) A__IF(true, if(div(y0, y1), y2, y3), z1) -> c10(MARK(if(div(y0, y1), y2, y3))) A__IF(true, if(if(minus(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(geq(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(div(y0, y1), y2, y3), y4, y5), z1) -> c10(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(true, if(if(if(y0, y1, y2), y3, y4), y5, y6), z1) -> c10(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(true, if(if(s(y0), y1, y2), y3, y4), z1) -> c10(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(true, if(if(true, y0, y1), y2, y3), z1) -> c10(MARK(if(if(true, y0, y1), y2, y3))) A__IF(true, if(if(false, y0, y1), y2, y3), z1) -> c10(MARK(if(if(false, y0, y1), y2, y3))) A__IF(true, if(if(y0, y1, y2), y3, y4), z1) -> c10(MARK(if(if(y0, y1, y2), y3, y4))) A__IF(false, z0, minus(y0, y1)) -> c11(MARK(minus(y0, y1))) A__IF(false, z0, geq(y0, y1)) -> c11(MARK(geq(y0, y1))) A__IF(false, z0, s(y0)) -> c11(MARK(s(y0))) A__IF(false, z0, div(s(y0), y1)) -> c11(MARK(div(s(y0), y1))) A__IF(false, z0, if(s(y0), y1, y2)) -> c11(MARK(if(s(y0), y1, y2))) A__IF(false, z0, if(true, y0, y1)) -> c11(MARK(if(true, y0, y1))) A__IF(false, z0, if(false, y0, y1)) -> c11(MARK(if(false, y0, y1))) A__IF(false, z0, div(minus(s(y0), s(y1)), y2)) -> c11(MARK(div(minus(s(y0), s(y1)), y2))) A__IF(false, z0, div(minus(y0, y1), y2)) -> c11(MARK(div(minus(y0, y1), y2))) A__IF(false, z0, div(geq(s(y0), s(y1)), y2)) -> c11(MARK(div(geq(s(y0), s(y1)), y2))) A__IF(false, z0, div(geq(y0, y1), y2)) -> c11(MARK(div(geq(y0, y1), y2))) A__IF(false, z0, div(div(minus(y0, y1), y2), y3)) -> c11(MARK(div(div(minus(y0, y1), y2), y3))) A__IF(false, z0, div(div(geq(y0, y1), y2), y3)) -> c11(MARK(div(div(geq(y0, y1), y2), y3))) A__IF(false, z0, div(div(div(y0, y1), y2), y3)) -> c11(MARK(div(div(div(y0, y1), y2), y3))) A__IF(false, z0, div(div(if(y0, y1, y2), y3), y4)) -> c11(MARK(div(div(if(y0, y1, y2), y3), y4))) A__IF(false, z0, div(div(s(y0), y1), y2)) -> c11(MARK(div(div(s(y0), y1), y2))) A__IF(false, z0, div(div(y0, y1), y2)) -> c11(MARK(div(div(y0, y1), y2))) A__IF(false, z0, div(if(minus(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(minus(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(geq(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(geq(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(div(y0, y1), y2, y3), y4)) -> c11(MARK(div(if(div(y0, y1), y2, y3), y4))) A__IF(false, z0, div(if(if(y0, y1, y2), y3, y4), y5)) -> c11(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) A__IF(false, z0, div(if(s(y0), y1, y2), y3)) -> c11(MARK(div(if(s(y0), y1, y2), y3))) A__IF(false, z0, div(if(true, y0, y1), y2)) -> c11(MARK(div(if(true, y0, y1), y2))) A__IF(false, z0, div(if(false, y0, y1), y2)) -> c11(MARK(div(if(false, y0, y1), y2))) A__IF(false, z0, div(if(y0, y1, y2), y3)) -> c11(MARK(div(if(y0, y1, y2), y3))) A__IF(false, z0, if(minus(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(minus(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(minus(y0, y1), y2, y3)) -> c11(MARK(if(minus(y0, y1), y2, y3))) A__IF(false, z0, if(geq(y0, 0), y1, y2)) -> c11(MARK(if(geq(y0, 0), y1, y2))) A__IF(false, z0, if(geq(0, s(y0)), y1, y2)) -> c11(MARK(if(geq(0, s(y0)), y1, y2))) A__IF(false, z0, if(geq(s(y0), s(y1)), y2, y3)) -> c11(MARK(if(geq(s(y0), s(y1)), y2, y3))) A__IF(false, z0, if(geq(y0, y1), y2, y3)) -> c11(MARK(if(geq(y0, y1), y2, y3))) A__IF(false, z0, if(div(minus(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(minus(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(geq(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(geq(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(div(y0, y1), y2), y3, y4)) -> c11(MARK(if(div(div(y0, y1), y2), y3, y4))) A__IF(false, z0, if(div(if(y0, y1, y2), y3), y4, y5)) -> c11(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) A__IF(false, z0, if(div(s(y0), y1), y2, y3)) -> c11(MARK(if(div(s(y0), y1), y2, y3))) A__IF(false, z0, if(div(y0, y1), y2, y3)) -> c11(MARK(if(div(y0, y1), y2, y3))) A__IF(false, z0, if(if(minus(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(geq(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(div(y0, y1), y2, y3), y4, y5)) -> c11(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) A__IF(false, z0, if(if(if(y0, y1, y2), y3, y4), y5, y6)) -> c11(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) A__IF(false, z0, if(if(s(y0), y1, y2), y3, y4)) -> c11(MARK(if(if(s(y0), y1, y2), y3, y4))) A__IF(false, z0, if(if(true, y0, y1), y2, y3)) -> c11(MARK(if(if(true, y0, y1), y2, y3))) A__IF(false, z0, if(if(false, y0, y1), y2, y3)) -> c11(MARK(if(if(false, y0, y1), y2, y3))) A__IF(false, z0, if(if(y0, y1, y2), y3, y4)) -> c11(MARK(if(if(y0, y1, y2), y3, y4))) MARK(minus(s(s(y0)), s(s(y1)))) -> c13(A__MINUS(s(s(y0)), s(s(y1)))) MARK(s(s(y0))) -> c18(MARK(s(y0))) MARK(s(div(s(y0), y1))) -> c18(MARK(div(s(y0), y1))) MARK(s(if(s(y0), y1, y2))) -> c18(MARK(if(s(y0), y1, y2))) MARK(s(if(true, y0, y1))) -> c18(MARK(if(true, y0, y1))) MARK(s(if(false, y0, y1))) -> c18(MARK(if(false, y0, y1))) MARK(s(div(minus(s(y0), s(y1)), y2))) -> c18(MARK(div(minus(s(y0), s(y1)), y2))) MARK(s(div(minus(y0, y1), y2))) -> c18(MARK(div(minus(y0, y1), y2))) MARK(s(div(geq(s(y0), s(y1)), y2))) -> c18(MARK(div(geq(s(y0), s(y1)), y2))) MARK(s(div(geq(y0, y1), y2))) -> c18(MARK(div(geq(y0, y1), y2))) MARK(s(div(div(minus(y0, y1), y2), y3))) -> c18(MARK(div(div(minus(y0, y1), y2), y3))) MARK(s(div(div(geq(y0, y1), y2), y3))) -> c18(MARK(div(div(geq(y0, y1), y2), y3))) MARK(s(div(div(div(y0, y1), y2), y3))) -> c18(MARK(div(div(div(y0, y1), y2), y3))) MARK(s(div(div(if(y0, y1, y2), y3), y4))) -> c18(MARK(div(div(if(y0, y1, y2), y3), y4))) MARK(s(div(div(s(y0), y1), y2))) -> c18(MARK(div(div(s(y0), y1), y2))) MARK(s(div(div(y0, y1), y2))) -> c18(MARK(div(div(y0, y1), y2))) MARK(s(div(if(minus(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(minus(y0, y1), y2, y3), y4))) MARK(s(div(if(geq(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(geq(y0, y1), y2, y3), y4))) MARK(s(div(if(div(y0, y1), y2, y3), y4))) -> c18(MARK(div(if(div(y0, y1), y2, y3), y4))) MARK(s(div(if(if(y0, y1, y2), y3, y4), y5))) -> c18(MARK(div(if(if(y0, y1, y2), y3, y4), y5))) MARK(s(div(if(s(y0), y1, y2), y3))) -> c18(MARK(div(if(s(y0), y1, y2), y3))) MARK(s(div(if(true, y0, y1), y2))) -> c18(MARK(div(if(true, y0, y1), y2))) MARK(s(div(if(false, y0, y1), y2))) -> c18(MARK(div(if(false, y0, y1), y2))) MARK(s(div(if(y0, y1, y2), y3))) -> c18(MARK(div(if(y0, y1, y2), y3))) MARK(s(if(minus(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(minus(s(y0), s(y1)), y2, y3))) MARK(s(if(minus(y0, y1), y2, y3))) -> c18(MARK(if(minus(y0, y1), y2, y3))) MARK(s(if(geq(s(y0), s(y1)), y2, y3))) -> c18(MARK(if(geq(s(y0), s(y1)), y2, y3))) MARK(s(if(geq(y0, y1), y2, y3))) -> c18(MARK(if(geq(y0, y1), y2, y3))) MARK(s(if(div(minus(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(minus(y0, y1), y2), y3, y4))) MARK(s(if(div(geq(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(geq(y0, y1), y2), y3, y4))) MARK(s(if(div(div(y0, y1), y2), y3, y4))) -> c18(MARK(if(div(div(y0, y1), y2), y3, y4))) MARK(s(if(div(if(y0, y1, y2), y3), y4, y5))) -> c18(MARK(if(div(if(y0, y1, y2), y3), y4, y5))) MARK(s(if(div(s(y0), y1), y2, y3))) -> c18(MARK(if(div(s(y0), y1), y2, y3))) MARK(s(if(div(y0, y1), y2, y3))) -> c18(MARK(if(div(y0, y1), y2, y3))) MARK(s(if(if(minus(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(minus(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(geq(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(geq(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(div(y0, y1), y2, y3), y4, y5))) -> c18(MARK(if(if(div(y0, y1), y2, y3), y4, y5))) MARK(s(if(if(if(y0, y1, y2), y3, y4), y5, y6))) -> c18(MARK(if(if(if(y0, y1, y2), y3, y4), y5, y6))) MARK(s(if(if(s(y0), y1, y2), y3, y4))) -> c18(MARK(if(if(s(y0), y1, y2), y3, y4))) MARK(s(if(if(true, y0, y1), y2, y3))) -> c18(MARK(if(if(true, y0, y1), y2, y3))) MARK(s(if(if(false, y0, y1), y2, y3))) -> c18(MARK(if(if(false, y0, y1), y2, y3))) MARK(s(if(if(y0, y1, y2), y3, y4))) -> c18(MARK(if(if(y0, y1, y2), y3, y4))) MARK(s(minus(s(s(y0)), s(s(y1))))) -> c18(MARK(minus(s(s(y0)), s(s(y1))))) MARK(s(geq(s(s(y0)), s(s(y1))))) -> c18(MARK(geq(s(s(y0)), s(s(y1))))) MARK(s(if(geq(y0, 0), y1, y2))) -> c18(MARK(if(geq(y0, 0), y1, y2))) MARK(s(if(geq(0, s(y0)), y1, y2))) -> c18(MARK(if(geq(0, s(y0)), y1, y2))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(A__DIV(a__minus(s(y0), s(y1)), z2), MARK(minus(s(s(y0)), s(s(y1))))) K tuples: MARK(geq(s(s(y0)), s(s(y1)))) -> c14(A__GEQ(s(s(y0)), s(s(y1)))) A__DIV(s(s(s(y0))), s(s(s(y1)))) -> c8(A__GEQ(s(s(y0)), s(s(y1)))) MARK(div(minus(s(s(y0)), s(s(y1))), z2)) -> c15(MARK(minus(s(s(y0)), s(s(y1))))) Defined Rule Symbols: a__minus_2, a__geq_2, a__div_2, a__if_3, mark_1 Defined Pair Symbols: A__DIV_2, MARK_1, A__MINUS_2, A__GEQ_2, A__IF_3 Compound Symbols: c8_1, c15_2, c8_2, c16_2, c16_1, c1_1, c5_1, c10_1, c11_1, c13_1, c14_1, c18_1, c15_1 ---------------------------------------- (225) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (226) 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: a__minus(0, Y) -> 0 [1] a__minus(s(X), s(Y)) -> a__minus(X, Y) [1] a__geq(X, 0) -> true [1] a__geq(0, s(Y)) -> false [1] a__geq(s(X), s(Y)) -> a__geq(X, Y) [1] a__div(0, s(Y)) -> 0 [1] a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1] a__if(true, X, Y) -> mark(X) [1] a__if(false, X, Y) -> mark(Y) [1] mark(minus(X1, X2)) -> a__minus(X1, X2) [1] mark(geq(X1, X2)) -> a__geq(X1, X2) [1] mark(div(X1, X2)) -> a__div(mark(X1), X2) [1] mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) [1] mark(0) -> 0 [1] mark(s(X)) -> s(mark(X)) [1] mark(true) -> true [1] mark(false) -> false [1] a__minus(X1, X2) -> minus(X1, X2) [1] a__geq(X1, X2) -> geq(X1, X2) [1] a__div(X1, X2) -> div(X1, X2) [1] a__if(X1, X2, X3) -> if(X1, X2, X3) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (227) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (228) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a__minus(0, Y) -> 0 [1] a__minus(s(X), s(Y)) -> a__minus(X, Y) [1] a__geq(X, 0) -> true [1] a__geq(0, s(Y)) -> false [1] a__geq(s(X), s(Y)) -> a__geq(X, Y) [1] a__div(0, s(Y)) -> 0 [1] a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1] a__if(true, X, Y) -> mark(X) [1] a__if(false, X, Y) -> mark(Y) [1] mark(minus(X1, X2)) -> a__minus(X1, X2) [1] mark(geq(X1, X2)) -> a__geq(X1, X2) [1] mark(div(X1, X2)) -> a__div(mark(X1), X2) [1] mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) [1] mark(0) -> 0 [1] mark(s(X)) -> s(mark(X)) [1] mark(true) -> true [1] mark(false) -> false [1] a__minus(X1, X2) -> minus(X1, X2) [1] a__geq(X1, X2) -> geq(X1, X2) [1] a__div(X1, X2) -> div(X1, X2) [1] a__if(X1, X2, X3) -> if(X1, X2, X3) [1] The TRS has the following type information: a__minus :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if 0 :: 0:s:true:false:minus:div:geq:if s :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if a__geq :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if true :: 0:s:true:false:minus:div:geq:if false :: 0:s:true:false:minus:div:geq:if a__div :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if a__if :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if div :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if minus :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if mark :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if geq :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if if :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if Rewrite Strategy: INNERMOST ---------------------------------------- (229) 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: none (c) The following functions are completely defined: mark_1 a__geq_2 a__if_3 a__minus_2 a__div_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (230) 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: a__minus(0, Y) -> 0 [1] a__minus(s(X), s(Y)) -> a__minus(X, Y) [1] a__geq(X, 0) -> true [1] a__geq(0, s(Y)) -> false [1] a__geq(s(X), s(Y)) -> a__geq(X, Y) [1] a__div(0, s(Y)) -> 0 [1] a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1] a__if(true, X, Y) -> mark(X) [1] a__if(false, X, Y) -> mark(Y) [1] mark(minus(X1, X2)) -> a__minus(X1, X2) [1] mark(geq(X1, X2)) -> a__geq(X1, X2) [1] mark(div(X1, X2)) -> a__div(mark(X1), X2) [1] mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) [1] mark(0) -> 0 [1] mark(s(X)) -> s(mark(X)) [1] mark(true) -> true [1] mark(false) -> false [1] a__minus(X1, X2) -> minus(X1, X2) [1] a__geq(X1, X2) -> geq(X1, X2) [1] a__div(X1, X2) -> div(X1, X2) [1] a__if(X1, X2, X3) -> if(X1, X2, X3) [1] The TRS has the following type information: a__minus :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if 0 :: 0:s:true:false:minus:div:geq:if s :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if a__geq :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if true :: 0:s:true:false:minus:div:geq:if false :: 0:s:true:false:minus:div:geq:if a__div :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if a__if :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if div :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if minus :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if mark :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if geq :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if if :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if Rewrite Strategy: INNERMOST ---------------------------------------- (231) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (232) 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: a__minus(0, Y) -> 0 [1] a__minus(s(X), s(Y)) -> a__minus(X, Y) [1] a__geq(X, 0) -> true [1] a__geq(0, s(Y)) -> false [1] a__geq(s(X), s(Y)) -> a__geq(X, Y) [1] a__div(0, s(Y)) -> 0 [1] a__div(s(X), s(0)) -> a__if(true, s(div(minus(X, 0), s(0))), 0) [2] a__div(s(0), s(s(Y'))) -> a__if(false, s(div(minus(0, s(Y')), s(s(Y')))), 0) [2] a__div(s(s(X')), s(s(Y''))) -> a__if(a__geq(X', Y''), s(div(minus(s(X'), s(Y'')), s(s(Y'')))), 0) [2] a__div(s(X), s(Y)) -> a__if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [2] a__if(true, X, Y) -> mark(X) [1] a__if(false, X, Y) -> mark(Y) [1] mark(minus(X1, X2)) -> a__minus(X1, X2) [1] mark(geq(X1, X2)) -> a__geq(X1, X2) [1] mark(div(minus(X1', X2'), X2)) -> a__div(a__minus(X1', X2'), X2) [2] mark(div(geq(X1'', X2''), X2)) -> a__div(a__geq(X1'', X2''), X2) [2] mark(div(div(X11, X21), X2)) -> a__div(a__div(mark(X11), X21), X2) [2] mark(div(if(X12, X22, X3'), X2)) -> a__div(a__if(mark(X12), X22, X3'), X2) [2] mark(div(0, X2)) -> a__div(0, X2) [2] mark(div(s(X''), X2)) -> a__div(s(mark(X'')), X2) [2] mark(div(true, X2)) -> a__div(true, X2) [2] mark(div(false, X2)) -> a__div(false, X2) [2] mark(if(minus(X13, X23), X2, X3)) -> a__if(a__minus(X13, X23), X2, X3) [2] mark(if(geq(X14, X24), X2, X3)) -> a__if(a__geq(X14, X24), X2, X3) [2] mark(if(div(X15, X25), X2, X3)) -> a__if(a__div(mark(X15), X25), X2, X3) [2] mark(if(if(X16, X26, X3''), X2, X3)) -> a__if(a__if(mark(X16), X26, X3''), X2, X3) [2] mark(if(0, X2, X3)) -> a__if(0, X2, X3) [2] mark(if(s(X4), X2, X3)) -> a__if(s(mark(X4)), X2, X3) [2] mark(if(true, X2, X3)) -> a__if(true, X2, X3) [2] mark(if(false, X2, X3)) -> a__if(false, X2, X3) [2] mark(0) -> 0 [1] mark(s(X)) -> s(mark(X)) [1] mark(true) -> true [1] mark(false) -> false [1] a__minus(X1, X2) -> minus(X1, X2) [1] a__geq(X1, X2) -> geq(X1, X2) [1] a__div(X1, X2) -> div(X1, X2) [1] a__if(X1, X2, X3) -> if(X1, X2, X3) [1] The TRS has the following type information: a__minus :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if 0 :: 0:s:true:false:minus:div:geq:if s :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if a__geq :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if true :: 0:s:true:false:minus:div:geq:if false :: 0:s:true:false:minus:div:geq:if a__div :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if a__if :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if div :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if minus :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if mark :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if geq :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if if :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if Rewrite Strategy: INNERMOST ---------------------------------------- (233) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 false => 1 ---------------------------------------- (234) Obligation: Complexity RNTS consisting of the following rules: a__div(z, z') -{ 2 }-> a__if(a__geq(X', Y''), 1 + (1 + (1 + (1 + X') + (1 + Y'')) + (1 + (1 + Y''))), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X') a__div(z, z') -{ 2 }-> a__if(2, 1 + (1 + (1 + X + 0) + (1 + 0)), 0) :|: z = 1 + X, X >= 0, z' = 1 + 0 a__div(z, z') -{ 2 }-> a__if(1, 1 + (1 + (1 + 0 + (1 + Y')) + (1 + (1 + Y'))), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0 a__div(z, z') -{ 2 }-> a__if(1 + X + Y, 1 + (1 + (1 + X + Y) + (1 + Y)), 0) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 a__div(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0 a__div(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__geq(z, z') -{ 1 }-> a__geq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 a__geq(z, z') -{ 1 }-> 2 :|: X >= 0, z = X, z' = 0 a__geq(z, z') -{ 1 }-> 1 :|: Y >= 0, z' = 1 + Y, z = 0 a__geq(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__if(z, z', z'') -{ 1 }-> mark(X) :|: z = 2, z' = X, Y >= 0, z'' = Y, X >= 0 a__if(z, z', z'') -{ 1 }-> mark(Y) :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0 a__if(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3 a__minus(z, z') -{ 1 }-> a__minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 a__minus(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0 a__minus(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 mark(z) -{ 1 }-> a__minus(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 2 }-> a__if(a__minus(X13, X23), X2, X3) :|: X3 >= 0, X2 >= 0, X13 >= 0, X23 >= 0, z = 1 + (1 + X13 + X23) + X2 + X3 mark(z) -{ 2 }-> a__if(a__if(mark(X16), X26, X3''), X2, X3) :|: z = 1 + (1 + X16 + X26 + X3'') + X2 + X3, X16 >= 0, X26 >= 0, X3'' >= 0, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(a__geq(X14, X24), X2, X3) :|: X3 >= 0, X2 >= 0, z = 1 + (1 + X14 + X24) + X2 + X3, X14 >= 0, X24 >= 0 mark(z) -{ 2 }-> a__if(a__div(mark(X15), X25), X2, X3) :|: z = 1 + (1 + X15 + X25) + X2 + X3, X3 >= 0, X2 >= 0, X15 >= 0, X25 >= 0 mark(z) -{ 2 }-> a__if(2, X2, X3) :|: z = 1 + 2 + X2 + X3, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(1, X2, X3) :|: z = 1 + 1 + X2 + X3, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(0, X2, X3) :|: z = 1 + 0 + X2 + X3, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(1 + mark(X4), X2, X3) :|: z = 1 + (1 + X4) + X2 + X3, X3 >= 0, X4 >= 0, X2 >= 0 mark(z) -{ 1 }-> a__geq(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 2 }-> a__div(a__minus(X1', X2'), X2) :|: X2' >= 0, X1' >= 0, X2 >= 0, z = 1 + (1 + X1' + X2') + X2 mark(z) -{ 2 }-> a__div(a__if(mark(X12), X22, X3'), X2) :|: X3' >= 0, X12 >= 0, X22 >= 0, z = 1 + (1 + X12 + X22 + X3') + X2, X2 >= 0 mark(z) -{ 2 }-> a__div(a__geq(X1'', X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0 mark(z) -{ 2 }-> a__div(a__div(mark(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__div(2, X2) :|: z = 1 + 2 + X2, X2 >= 0 mark(z) -{ 2 }-> a__div(1, X2) :|: z = 1 + 1 + X2, X2 >= 0 mark(z) -{ 2 }-> a__div(0, X2) :|: z = 1 + 0 + X2, X2 >= 0 mark(z) -{ 2 }-> a__div(1 + mark(X''), X2) :|: z = 1 + (1 + X'') + X2, X'' >= 0, X2 >= 0 mark(z) -{ 1 }-> 2 :|: z = 2 mark(z) -{ 1 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 1 }-> 1 + mark(X) :|: z = 1 + X, X >= 0 ---------------------------------------- (235) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (236) Obligation: Complexity RNTS consisting of the following rules: a__div(z, z') -{ 2 }-> a__if(a__geq(z - 2, z' - 2), 1 + (1 + (1 + (1 + (z - 2)) + (1 + (z' - 2))) + (1 + (1 + (z' - 2)))), 0) :|: z' - 2 >= 0, z - 2 >= 0 a__div(z, z') -{ 2 }-> a__if(2, 1 + (1 + (1 + (z - 1) + 0) + (1 + 0)), 0) :|: z - 1 >= 0, z' = 1 + 0 a__div(z, z') -{ 2 }-> a__if(1, 1 + (1 + (1 + 0 + (1 + (z' - 2))) + (1 + (1 + (z' - 2)))), 0) :|: z' - 2 >= 0, z = 1 + 0 a__div(z, z') -{ 2 }-> a__if(1 + (z - 1) + (z' - 1), 1 + (1 + (1 + (z - 1) + (z' - 1)) + (1 + (z' - 1))), 0) :|: z' - 1 >= 0, z - 1 >= 0 a__div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 a__div(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 a__geq(z, z') -{ 1 }-> a__geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 a__geq(z, z') -{ 1 }-> 2 :|: z >= 0, z' = 0 a__geq(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0 a__geq(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 a__if(z, z', z'') -{ 1 }-> mark(z') :|: z = 2, z'' >= 0, z' >= 0 a__if(z, z', z'') -{ 1 }-> mark(z'') :|: z'' >= 0, z = 1, z' >= 0 a__if(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0 a__minus(z, z') -{ 1 }-> a__minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 a__minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 a__minus(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 mark(z) -{ 1 }-> a__minus(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 2 }-> a__if(a__minus(X13, X23), X2, X3) :|: X3 >= 0, X2 >= 0, X13 >= 0, X23 >= 0, z = 1 + (1 + X13 + X23) + X2 + X3 mark(z) -{ 2 }-> a__if(a__if(mark(X16), X26, X3''), X2, X3) :|: z = 1 + (1 + X16 + X26 + X3'') + X2 + X3, X16 >= 0, X26 >= 0, X3'' >= 0, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(a__geq(X14, X24), X2, X3) :|: X3 >= 0, X2 >= 0, z = 1 + (1 + X14 + X24) + X2 + X3, X14 >= 0, X24 >= 0 mark(z) -{ 2 }-> a__if(a__div(mark(X15), X25), X2, X3) :|: z = 1 + (1 + X15 + X25) + X2 + X3, X3 >= 0, X2 >= 0, X15 >= 0, X25 >= 0 mark(z) -{ 2 }-> a__if(2, X2, X3) :|: z = 1 + 2 + X2 + X3, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(1, X2, X3) :|: z = 1 + 1 + X2 + X3, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(0, X2, X3) :|: z = 1 + 0 + X2 + X3, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(1 + mark(X4), X2, X3) :|: z = 1 + (1 + X4) + X2 + X3, X3 >= 0, X4 >= 0, X2 >= 0 mark(z) -{ 1 }-> a__geq(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 2 }-> a__div(a__minus(X1', X2'), X2) :|: X2' >= 0, X1' >= 0, X2 >= 0, z = 1 + (1 + X1' + X2') + X2 mark(z) -{ 2 }-> a__div(a__if(mark(X12), X22, X3'), X2) :|: X3' >= 0, X12 >= 0, X22 >= 0, z = 1 + (1 + X12 + X22 + X3') + X2, X2 >= 0 mark(z) -{ 2 }-> a__div(a__geq(X1'', X2''), X2) :|: X1'' >= 0, z = 1 + (1 + X1'' + X2'') + X2, X2'' >= 0, X2 >= 0 mark(z) -{ 2 }-> a__div(a__div(mark(X11), X21), X2) :|: z = 1 + (1 + X11 + X21) + X2, X11 >= 0, X21 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__div(2, z - 3) :|: z - 3 >= 0 mark(z) -{ 2 }-> a__div(1, z - 2) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__div(0, z - 1) :|: z - 1 >= 0 mark(z) -{ 2 }-> a__div(1 + mark(X''), X2) :|: z = 1 + (1 + X'') + X2, X'' >= 0, X2 >= 0 mark(z) -{ 1 }-> 2 :|: z = 2 mark(z) -{ 1 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 1 >= 0 ---------------------------------------- (237) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (238) 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: a__minus(0, Y) -> 0 [1] a__minus(s(X), s(Y)) -> a__minus(X, Y) [1] a__geq(X, 0) -> true [1] a__geq(0, s(Y)) -> false [1] a__geq(s(X), s(Y)) -> a__geq(X, Y) [1] a__div(0, s(Y)) -> 0 [1] a__div(s(X), s(Y)) -> a__if(a__geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1] a__if(true, X, Y) -> mark(X) [1] a__if(false, X, Y) -> mark(Y) [1] mark(minus(X1, X2)) -> a__minus(X1, X2) [1] mark(geq(X1, X2)) -> a__geq(X1, X2) [1] mark(div(X1, X2)) -> a__div(mark(X1), X2) [1] mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) [1] mark(0) -> 0 [1] mark(s(X)) -> s(mark(X)) [1] mark(true) -> true [1] mark(false) -> false [1] a__minus(X1, X2) -> minus(X1, X2) [1] a__geq(X1, X2) -> geq(X1, X2) [1] a__div(X1, X2) -> div(X1, X2) [1] a__if(X1, X2, X3) -> if(X1, X2, X3) [1] The TRS has the following type information: a__minus :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if 0 :: 0:s:true:false:minus:div:geq:if s :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if a__geq :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if true :: 0:s:true:false:minus:div:geq:if false :: 0:s:true:false:minus:div:geq:if a__div :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if a__if :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if div :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if minus :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if mark :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if geq :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if if :: 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if -> 0:s:true:false:minus:div:geq:if Rewrite Strategy: INNERMOST ---------------------------------------- (239) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 false => 1 ---------------------------------------- (240) Obligation: Complexity RNTS consisting of the following rules: a__div(z, z') -{ 1 }-> a__if(a__geq(X, Y), 1 + (1 + (1 + X + Y) + (1 + Y)), 0) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 a__div(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0 a__div(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__geq(z, z') -{ 1 }-> a__geq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 a__geq(z, z') -{ 1 }-> 2 :|: X >= 0, z = X, z' = 0 a__geq(z, z') -{ 1 }-> 1 :|: Y >= 0, z' = 1 + Y, z = 0 a__geq(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__if(z, z', z'') -{ 1 }-> mark(X) :|: z = 2, z' = X, Y >= 0, z'' = Y, X >= 0 a__if(z, z', z'') -{ 1 }-> mark(Y) :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0 a__if(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3 a__minus(z, z') -{ 1 }-> a__minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0 a__minus(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0 a__minus(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 mark(z) -{ 1 }-> a__minus(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 1 }-> a__if(mark(X1), X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0 mark(z) -{ 1 }-> a__geq(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 1 }-> a__div(mark(X1), X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 1 }-> 2 :|: z = 2 mark(z) -{ 1 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 1 }-> 1 + mark(X) :|: z = 1 + X, X >= 0 Only complete derivations are relevant for the runtime complexity.