KILLED proof of input_6SZ76lI23S.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CompletionProof [UPPER BOUND(ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (22) CdtProblem (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 184 ms] (26) CdtProblem (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 37 ms] (32) CdtProblem (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 696 ms] (38) CdtProblem (39) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 617 ms] (40) CdtProblem (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 55 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 54 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 23 ms] (88) CdtProblem (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms] (92) CdtProblem (93) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 47 ms] (100) CdtProblem (101) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 103 ms] (104) CdtProblem (105) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 83 ms] (108) CdtProblem (109) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 35 ms] (112) CdtProblem (113) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 131 ms] (116) CdtProblem (117) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 32 ms] (128) CdtProblem (129) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 317 ms] (132) CdtProblem (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 2 ms] (136) CdtProblem (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 4 ms] (148) CdtProblem (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 18 ms] (150) CdtProblem (151) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 26 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 4 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 784 ms] (162) CdtProblem (163) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 726 ms] (172) CdtProblem (173) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 13 ms] (174) CdtProblem (175) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 269 ms] (178) CdtProblem (179) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 13 ms] (180) CdtProblem (181) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 2 ms] (184) CdtProblem (185) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 1283 ms] (186) CdtProblem (187) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 4 ms] (190) CdtProblem (191) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 452 ms] (192) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: 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) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: 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 ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) 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 ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: 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 ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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 ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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 ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 false => 1 ---------------------------------------- (14) 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 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) 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 ---------------------------------------- (17) 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 ---------------------------------------- (18) 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 ---------------------------------------- (19) 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 ---------------------------------------- (20) 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. ---------------------------------------- (21) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (22) 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 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: MARK(0) -> c17 A__GEQ(z0, z1) -> c6 A__GEQ(z0, 0) -> c3 A__MINUS(z0, z1) -> c2 A__DIV(0, s(z0)) -> c7 A__GEQ(0, s(z0)) -> c4 MARK(false) -> c20 A__DIV(z0, z1) -> c9 A__IF(z0, z1, z2) -> c12 MARK(true) -> c19 A__MINUS(0, z0) -> c ---------------------------------------- (24) 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 ---------------------------------------- (25) 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)) 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_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] ---------------------------------------- (26) 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(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) 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 ---------------------------------------- (27) 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)) ---------------------------------------- (28) 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(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) 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__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 ---------------------------------------- (29) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (30) 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(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) 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__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 ---------------------------------------- (31) 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] ---------------------------------------- (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(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(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) 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 ---------------------------------------- (33) 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)) ---------------------------------------- (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__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(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) 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 ---------------------------------------- (35) 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)) ---------------------------------------- (36) 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(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) 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 ---------------------------------------- (37) 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 ---------------------------------------- (38) 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(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) 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 ---------------------------------------- (39) 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 ---------------------------------------- (40) 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(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) 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 ---------------------------------------- (41) 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)) ---------------------------------------- (42) 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(geq(z0, z1)) -> c14(A__GEQ(z0, z1)) MARK(if(z0, z1, z2)) -> c16(A__IF(mark(z0), z1, z2), MARK(z0)) 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 ---------------------------------------- (43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(if(0, x1, x2)) -> c16(A__IF(0, x1, x2), MARK(0)) ---------------------------------------- (44) 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 ---------------------------------------- (45) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (46) 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 ---------------------------------------- (47) 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))) ---------------------------------------- (48) 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 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (50) 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 ---------------------------------------- (51) 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 ---------------------------------------- (52) 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 ---------------------------------------- (53) 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))) ---------------------------------------- (54) 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 ---------------------------------------- (55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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__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 ---------------------------------------- (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(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 ---------------------------------------- (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__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 ---------------------------------------- (59) 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))) ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (63) 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))) ---------------------------------------- (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(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 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (67) 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))) ---------------------------------------- (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(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 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (71) 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))) ---------------------------------------- (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(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 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (75) 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))) ---------------------------------------- (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(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 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 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(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 ---------------------------------------- (79) 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))) ---------------------------------------- (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(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 ---------------------------------------- (81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 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(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 ---------------------------------------- (83) 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))) ---------------------------------------- (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(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 ---------------------------------------- (85) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (87) 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)) ---------------------------------------- (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(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 ---------------------------------------- (89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: 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)) ---------------------------------------- (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(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 ---------------------------------------- (91) 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)) ---------------------------------------- (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(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 ---------------------------------------- (93) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(div(div(true, z1), x1)) -> c15(A__DIV(div(true, z1), x1)) ---------------------------------------- (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(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 ---------------------------------------- (95) 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)) ---------------------------------------- (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(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 ---------------------------------------- (97) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(div(div(false, z1), x1)) -> c15(A__DIV(div(false, z1), x1)) ---------------------------------------- (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(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 ---------------------------------------- (99) 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)) ---------------------------------------- (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(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 ---------------------------------------- (101) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(div(if(0, z1, z2), x2)) -> c15(A__DIV(if(0, z1, z2), x2)) ---------------------------------------- (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(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 ---------------------------------------- (103) 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)) ---------------------------------------- (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(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 ---------------------------------------- (105) 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)) ---------------------------------------- (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(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 ---------------------------------------- (107) 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)) ---------------------------------------- (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(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 ---------------------------------------- (109) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(if(div(true, z1), x1, x2)) -> c16(A__IF(div(true, z1), x1, x2)) ---------------------------------------- (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(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 ---------------------------------------- (111) 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)) ---------------------------------------- (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(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 ---------------------------------------- (113) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(if(div(false, z1), x1, x2)) -> c16(A__IF(div(false, z1), x1, x2)) ---------------------------------------- (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(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 ---------------------------------------- (115) 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)) ---------------------------------------- (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(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 ---------------------------------------- (117) 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)) ---------------------------------------- (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(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 ---------------------------------------- (119) 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))) ---------------------------------------- (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__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 ---------------------------------------- (121) 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))) ---------------------------------------- (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__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 ---------------------------------------- (123) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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__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 ---------------------------------------- (125) 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))) ---------------------------------------- (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__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 ---------------------------------------- (127) 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))) ---------------------------------------- (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: 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 ---------------------------------------- (129) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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: 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 ---------------------------------------- (131) 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 ---------------------------------------- (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: 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 ---------------------------------------- (133) 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)))) ---------------------------------------- (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: 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 ---------------------------------------- (135) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: A__DIV(s(s(0)), s(s(s(z0)))) -> c8 A__DIV(s(0), s(s(z0))) -> c8 MARK(if(minus(0, z0), x2, x3)) -> c16(MARK(minus(0, z0))) MARK(div(minus(0, z0), x2)) -> c15(MARK(minus(0, z0))) ---------------------------------------- (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: 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 ---------------------------------------- (137) 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)))) ---------------------------------------- (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: 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 ---------------------------------------- (139) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(div(geq(0, s(z0)), x2)) -> c15(MARK(geq(0, s(z0)))) MARK(div(geq(z0, 0), x2)) -> c15(MARK(geq(z0, 0))) ---------------------------------------- (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: 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 ---------------------------------------- (141) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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: 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 ---------------------------------------- (143) 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))) ---------------------------------------- (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__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 ---------------------------------------- (145) 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)))) ---------------------------------------- (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__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 ---------------------------------------- (147) 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)))) ---------------------------------------- (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__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 ---------------------------------------- (149) 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)) ---------------------------------------- (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__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 ---------------------------------------- (151) 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))) ---------------------------------------- (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__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 ---------------------------------------- (153) 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)))) ---------------------------------------- (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__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 ---------------------------------------- (155) 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))))) ---------------------------------------- (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__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 ---------------------------------------- (157) 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))))) ---------------------------------------- (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__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 ---------------------------------------- (159) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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: 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 ---------------------------------------- (161) 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 ---------------------------------------- (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: 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 ---------------------------------------- (163) 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))))) ---------------------------------------- (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: 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 ---------------------------------------- (165) 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 ---------------------------------------- (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: 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 ---------------------------------------- (167) 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))))) ---------------------------------------- (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: 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 ---------------------------------------- (169) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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: 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 ---------------------------------------- (171) 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 ---------------------------------------- (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: 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 ---------------------------------------- (173) 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))) ---------------------------------------- (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: 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 ---------------------------------------- (175) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: 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 A__IF(true, div(geq(y0, 0), y1), z1) -> c10 ---------------------------------------- (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)) 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 ---------------------------------------- (177) 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)))) ---------------------------------------- (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))) 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 ---------------------------------------- (179) 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))))) ---------------------------------------- (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))) 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 ---------------------------------------- (181) 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))))) ---------------------------------------- (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))) 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 ---------------------------------------- (183) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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)))) 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 ---------------------------------------- (185) 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 ---------------------------------------- (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(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 ---------------------------------------- (187) 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))))) ---------------------------------------- (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(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 ---------------------------------------- (189) 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 ---------------------------------------- (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(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 ---------------------------------------- (191) 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))) ---------------------------------------- (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(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