KILLED proof of input_3CPqqSLkPI.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 10 ms] (14) CpxRNTS (15) InliningProof [UPPER BOUND(ID), 279 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CompletionProof [UPPER BOUND(ID), 0 ms] (20) CpxTypedWeightedCompleteTrs (21) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 15 ms] (22) CpxRNTS (23) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (24) CdtProblem (25) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 13 ms] (28) CdtProblem (29) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtLeafRemovalProof [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), 8 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), 6 ms] (84) CdtProblem (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 5 ms] (132) CdtProblem (133) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms] (178) CdtProblem (179) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 16 ms] (198) CdtProblem (199) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 195 ms] (212) CdtProblem (213) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 170 ms] (216) CdtProblem (217) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (218) CdtProblem (219) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 322 ms] (220) CdtProblem (221) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (222) 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__zeros -> cons(0, zeros) a__U11(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) 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__zeros -> cons(0', zeros) a__U11(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0') -> tt a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) a__length(nil) -> 0' a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0') -> 0' mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) 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__zeros -> cons(0, zeros) a__U11(tt, L) -> s(a__length(mark(L))) a__and(tt, X) -> mark(X) a__isNat(0) -> tt a__isNat(length(V1)) -> a__isNatList(V1) a__isNat(s(V1)) -> a__isNat(V1) a__isNatIList(V) -> a__isNatList(V) a__isNatIList(zeros) -> tt a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) a__isNatList(nil) -> tt a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(length(X)) -> a__length(mark(X)) mark(and(X1, X2)) -> a__and(mark(X1), X2) mark(isNat(X)) -> a__isNat(X) mark(isNatList(X)) -> a__isNatList(X) mark(isNatIList(X)) -> a__isNatIList(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(tt) -> tt mark(s(X)) -> s(mark(X)) mark(nil) -> nil a__zeros -> zeros a__U11(X1, X2) -> U11(X1, X2) a__length(X) -> length(X) a__and(X1, X2) -> and(X1, X2) a__isNat(X) -> isNat(X) a__isNatList(X) -> isNatList(X) a__isNatIList(X) -> isNatIList(X) 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__zeros -> cons(0, zeros) [1] a__U11(tt, L) -> s(a__length(mark(L))) [1] a__and(tt, X) -> mark(X) [1] a__isNat(0) -> tt [1] a__isNat(length(V1)) -> a__isNatList(V1) [1] a__isNat(s(V1)) -> a__isNat(V1) [1] a__isNatIList(V) -> a__isNatList(V) [1] a__isNatIList(zeros) -> tt [1] a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) [1] a__isNatList(nil) -> tt [1] a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) [1] a__length(nil) -> 0 [1] a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) [1] mark(zeros) -> a__zeros [1] mark(U11(X1, X2)) -> a__U11(mark(X1), X2) [1] mark(length(X)) -> a__length(mark(X)) [1] mark(and(X1, X2)) -> a__and(mark(X1), X2) [1] mark(isNat(X)) -> a__isNat(X) [1] mark(isNatList(X)) -> a__isNatList(X) [1] mark(isNatIList(X)) -> a__isNatIList(X) [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(0) -> 0 [1] mark(tt) -> tt [1] mark(s(X)) -> s(mark(X)) [1] mark(nil) -> nil [1] a__zeros -> zeros [1] a__U11(X1, X2) -> U11(X1, X2) [1] a__length(X) -> length(X) [1] a__and(X1, X2) -> and(X1, X2) [1] a__isNat(X) -> isNat(X) [1] a__isNatList(X) -> isNatList(X) [1] a__isNatIList(X) -> isNatIList(X) [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__zeros -> cons(0, zeros) [1] a__U11(tt, L) -> s(a__length(mark(L))) [1] a__and(tt, X) -> mark(X) [1] a__isNat(0) -> tt [1] a__isNat(length(V1)) -> a__isNatList(V1) [1] a__isNat(s(V1)) -> a__isNat(V1) [1] a__isNatIList(V) -> a__isNatList(V) [1] a__isNatIList(zeros) -> tt [1] a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) [1] a__isNatList(nil) -> tt [1] a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) [1] a__length(nil) -> 0 [1] a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) [1] mark(zeros) -> a__zeros [1] mark(U11(X1, X2)) -> a__U11(mark(X1), X2) [1] mark(length(X)) -> a__length(mark(X)) [1] mark(and(X1, X2)) -> a__and(mark(X1), X2) [1] mark(isNat(X)) -> a__isNat(X) [1] mark(isNatList(X)) -> a__isNatList(X) [1] mark(isNatIList(X)) -> a__isNatIList(X) [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(0) -> 0 [1] mark(tt) -> tt [1] mark(s(X)) -> s(mark(X)) [1] mark(nil) -> nil [1] a__zeros -> zeros [1] a__U11(X1, X2) -> U11(X1, X2) [1] a__length(X) -> length(X) [1] a__and(X1, X2) -> and(X1, X2) [1] a__isNat(X) -> isNat(X) [1] a__isNatList(X) -> isNatList(X) [1] a__isNatIList(X) -> isNatIList(X) [1] The TRS has the following type information: a__zeros :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and cons :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and 0 :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and zeros :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__U11 :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and tt :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and s :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__length :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and mark :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__and :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__isNat :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and length :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__isNatList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__isNatIList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and isNatIList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and nil :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and isNatList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and isNat :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and U11 :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and and :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and 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: a__and_2 a__isNatList_1 mark_1 a__isNat_1 a__length_1 a__U11_2 a__zeros a__isNatIList_1 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__zeros -> cons(0, zeros) [1] a__U11(tt, L) -> s(a__length(mark(L))) [1] a__and(tt, X) -> mark(X) [1] a__isNat(0) -> tt [1] a__isNat(length(V1)) -> a__isNatList(V1) [1] a__isNat(s(V1)) -> a__isNat(V1) [1] a__isNatIList(V) -> a__isNatList(V) [1] a__isNatIList(zeros) -> tt [1] a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) [1] a__isNatList(nil) -> tt [1] a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) [1] a__length(nil) -> 0 [1] a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) [1] mark(zeros) -> a__zeros [1] mark(U11(X1, X2)) -> a__U11(mark(X1), X2) [1] mark(length(X)) -> a__length(mark(X)) [1] mark(and(X1, X2)) -> a__and(mark(X1), X2) [1] mark(isNat(X)) -> a__isNat(X) [1] mark(isNatList(X)) -> a__isNatList(X) [1] mark(isNatIList(X)) -> a__isNatIList(X) [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(0) -> 0 [1] mark(tt) -> tt [1] mark(s(X)) -> s(mark(X)) [1] mark(nil) -> nil [1] a__zeros -> zeros [1] a__U11(X1, X2) -> U11(X1, X2) [1] a__length(X) -> length(X) [1] a__and(X1, X2) -> and(X1, X2) [1] a__isNat(X) -> isNat(X) [1] a__isNatList(X) -> isNatList(X) [1] a__isNatIList(X) -> isNatIList(X) [1] The TRS has the following type information: a__zeros :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and cons :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and 0 :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and zeros :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__U11 :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and tt :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and s :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__length :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and mark :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__and :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__isNat :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and length :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__isNatList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__isNatIList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and isNatIList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and nil :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and isNatList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and isNat :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and U11 :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and and :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and 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__zeros -> cons(0, zeros) [1] a__U11(tt, zeros) -> s(a__length(a__zeros)) [2] a__U11(tt, U11(X1', X2')) -> s(a__length(a__U11(mark(X1'), X2'))) [2] a__U11(tt, length(X')) -> s(a__length(a__length(mark(X')))) [2] a__U11(tt, and(X1'', X2'')) -> s(a__length(a__and(mark(X1''), X2''))) [2] a__U11(tt, isNat(X'')) -> s(a__length(a__isNat(X''))) [2] a__U11(tt, isNatList(X3)) -> s(a__length(a__isNatList(X3))) [2] a__U11(tt, isNatIList(X4)) -> s(a__length(a__isNatIList(X4))) [2] a__U11(tt, cons(X11, X21)) -> s(a__length(cons(mark(X11), X21))) [2] a__U11(tt, 0) -> s(a__length(0)) [2] a__U11(tt, tt) -> s(a__length(tt)) [2] a__U11(tt, s(X5)) -> s(a__length(s(mark(X5)))) [2] a__U11(tt, nil) -> s(a__length(nil)) [2] a__and(tt, X) -> mark(X) [1] a__isNat(0) -> tt [1] a__isNat(length(V1)) -> a__isNatList(V1) [1] a__isNat(s(V1)) -> a__isNat(V1) [1] a__isNatIList(V) -> a__isNatList(V) [1] a__isNatIList(zeros) -> tt [1] a__isNatIList(cons(0, V2)) -> a__and(tt, isNatIList(V2)) [2] a__isNatIList(cons(length(V1'), V2)) -> a__and(a__isNatList(V1'), isNatIList(V2)) [2] a__isNatIList(cons(s(V1''), V2)) -> a__and(a__isNat(V1''), isNatIList(V2)) [2] a__isNatIList(cons(V1, V2)) -> a__and(isNat(V1), isNatIList(V2)) [2] a__isNatList(nil) -> tt [1] a__isNatList(cons(0, V2)) -> a__and(tt, isNatList(V2)) [2] a__isNatList(cons(length(V11), V2)) -> a__and(a__isNatList(V11), isNatList(V2)) [2] a__isNatList(cons(s(V12), V2)) -> a__and(a__isNat(V12), isNatList(V2)) [2] a__isNatList(cons(V1, V2)) -> a__and(isNat(V1), isNatList(V2)) [2] a__length(nil) -> 0 [1] a__length(cons(N, nil)) -> a__U11(a__and(tt, isNat(N)), nil) [2] a__length(cons(N, cons(V13, V2'))) -> a__U11(a__and(a__and(a__isNat(V13), isNatList(V2')), isNat(N)), cons(V13, V2')) [2] a__length(cons(N, L)) -> a__U11(a__and(isNatList(L), isNat(N)), L) [2] mark(zeros) -> a__zeros [1] mark(U11(zeros, X2)) -> a__U11(a__zeros, X2) [2] mark(U11(U11(X12, X22), X2)) -> a__U11(a__U11(mark(X12), X22), X2) [2] mark(U11(length(X6), X2)) -> a__U11(a__length(mark(X6)), X2) [2] mark(U11(and(X13, X23), X2)) -> a__U11(a__and(mark(X13), X23), X2) [2] mark(U11(isNat(X7), X2)) -> a__U11(a__isNat(X7), X2) [2] mark(U11(isNatList(X8), X2)) -> a__U11(a__isNatList(X8), X2) [2] mark(U11(isNatIList(X9), X2)) -> a__U11(a__isNatIList(X9), X2) [2] mark(U11(cons(X14, X24), X2)) -> a__U11(cons(mark(X14), X24), X2) [2] mark(U11(0, X2)) -> a__U11(0, X2) [2] mark(U11(tt, X2)) -> a__U11(tt, X2) [2] mark(U11(s(X10), X2)) -> a__U11(s(mark(X10)), X2) [2] mark(U11(nil, X2)) -> a__U11(nil, X2) [2] mark(length(zeros)) -> a__length(a__zeros) [2] mark(length(U11(X15, X25))) -> a__length(a__U11(mark(X15), X25)) [2] mark(length(length(X16))) -> a__length(a__length(mark(X16))) [2] mark(length(and(X17, X26))) -> a__length(a__and(mark(X17), X26)) [2] mark(length(isNat(X18))) -> a__length(a__isNat(X18)) [2] mark(length(isNatList(X19))) -> a__length(a__isNatList(X19)) [2] mark(length(isNatIList(X20))) -> a__length(a__isNatIList(X20)) [2] mark(length(cons(X110, X27))) -> a__length(cons(mark(X110), X27)) [2] mark(length(0)) -> a__length(0) [2] mark(length(tt)) -> a__length(tt) [2] mark(length(s(X28))) -> a__length(s(mark(X28))) [2] mark(length(nil)) -> a__length(nil) [2] mark(and(zeros, X2)) -> a__and(a__zeros, X2) [2] mark(and(U11(X111, X29), X2)) -> a__and(a__U11(mark(X111), X29), X2) [2] mark(and(length(X30), X2)) -> a__and(a__length(mark(X30)), X2) [2] mark(and(and(X112, X210), X2)) -> a__and(a__and(mark(X112), X210), X2) [2] mark(and(isNat(X31), X2)) -> a__and(a__isNat(X31), X2) [2] mark(and(isNatList(X32), X2)) -> a__and(a__isNatList(X32), X2) [2] mark(and(isNatIList(X33), X2)) -> a__and(a__isNatIList(X33), X2) [2] mark(and(cons(X113, X211), X2)) -> a__and(cons(mark(X113), X211), X2) [2] mark(and(0, X2)) -> a__and(0, X2) [2] mark(and(tt, X2)) -> a__and(tt, X2) [2] mark(and(s(X34), X2)) -> a__and(s(mark(X34)), X2) [2] mark(and(nil, X2)) -> a__and(nil, X2) [2] mark(isNat(X)) -> a__isNat(X) [1] mark(isNatList(X)) -> a__isNatList(X) [1] mark(isNatIList(X)) -> a__isNatIList(X) [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(0) -> 0 [1] mark(tt) -> tt [1] mark(s(X)) -> s(mark(X)) [1] mark(nil) -> nil [1] a__zeros -> zeros [1] a__U11(X1, X2) -> U11(X1, X2) [1] a__length(X) -> length(X) [1] a__and(X1, X2) -> and(X1, X2) [1] a__isNat(X) -> isNat(X) [1] a__isNatList(X) -> isNatList(X) [1] a__isNatIList(X) -> isNatIList(X) [1] The TRS has the following type information: a__zeros :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and cons :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and 0 :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and zeros :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__U11 :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and tt :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and s :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__length :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and mark :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__and :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__isNat :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and length :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__isNatList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__isNatIList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and isNatIList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and nil :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and isNatList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and isNat :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and U11 :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and and :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and 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 zeros => 3 tt => 2 nil => 1 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: a__U11(z, z') -{ 2 }-> 1 + a__length(a__zeros) :|: z = 2, z' = 3 a__U11(z, z') -{ 2 }-> 1 + a__length(a__length(mark(X'))) :|: z = 2, X' >= 0, z' = 1 + X' a__U11(z, z') -{ 2 }-> 1 + a__length(a__isNatList(X3)) :|: z = 2, X3 >= 0, z' = 1 + X3 a__U11(z, z') -{ 2 }-> 1 + a__length(a__isNatIList(X4)) :|: z = 2, z' = 1 + X4, X4 >= 0 a__U11(z, z') -{ 2 }-> 1 + a__length(a__isNat(X'')) :|: z = 2, z' = 1 + X'', X'' >= 0 a__U11(z, z') -{ 2 }-> 1 + a__length(a__and(mark(X1''), X2'')) :|: z = 2, z' = 1 + X1'' + X2'', X1'' >= 0, X2'' >= 0 a__U11(z, z') -{ 2 }-> 1 + a__length(a__U11(mark(X1'), X2')) :|: z = 2, z' = 1 + X1' + X2', X2' >= 0, X1' >= 0 a__U11(z, z') -{ 2 }-> 1 + a__length(2) :|: z = 2, z' = 2 a__U11(z, z') -{ 2 }-> 1 + a__length(1) :|: z = 2, z' = 1 a__U11(z, z') -{ 2 }-> 1 + a__length(0) :|: z = 2, z' = 0 a__U11(z, z') -{ 2 }-> 1 + a__length(1 + mark(X5)) :|: z = 2, X5 >= 0, z' = 1 + X5 a__U11(z, z') -{ 2 }-> 1 + a__length(1 + mark(X11) + X21) :|: z = 2, z' = 1 + X11 + X21, X11 >= 0, X21 >= 0 a__U11(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__and(z, z') -{ 1 }-> mark(X) :|: z = 2, z' = X, X >= 0 a__and(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__isNat(z) -{ 1 }-> a__isNatList(V1) :|: z = 1 + V1, V1 >= 0 a__isNat(z) -{ 1 }-> a__isNat(V1) :|: z = 1 + V1, V1 >= 0 a__isNat(z) -{ 1 }-> 2 :|: z = 0 a__isNat(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__isNatIList(z) -{ 1 }-> a__isNatList(V) :|: z = V, V >= 0 a__isNatIList(z) -{ 2 }-> a__and(a__isNatList(V1'), 1 + V2) :|: V2 >= 0, z = 1 + (1 + V1') + V2, V1' >= 0 a__isNatIList(z) -{ 2 }-> a__and(a__isNat(V1''), 1 + V2) :|: V1'' >= 0, z = 1 + (1 + V1'') + V2, V2 >= 0 a__isNatIList(z) -{ 2 }-> a__and(2, 1 + V2) :|: V2 >= 0, z = 1 + 0 + V2 a__isNatIList(z) -{ 2 }-> a__and(1 + V1, 1 + V2) :|: V1 >= 0, V2 >= 0, z = 1 + V1 + V2 a__isNatIList(z) -{ 1 }-> 2 :|: z = 3 a__isNatIList(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__isNatList(z) -{ 2 }-> a__and(a__isNatList(V11), 1 + V2) :|: z = 1 + (1 + V11) + V2, V11 >= 0, V2 >= 0 a__isNatList(z) -{ 2 }-> a__and(a__isNat(V12), 1 + V2) :|: V12 >= 0, V2 >= 0, z = 1 + (1 + V12) + V2 a__isNatList(z) -{ 2 }-> a__and(2, 1 + V2) :|: V2 >= 0, z = 1 + 0 + V2 a__isNatList(z) -{ 2 }-> a__and(1 + V1, 1 + V2) :|: V1 >= 0, V2 >= 0, z = 1 + V1 + V2 a__isNatList(z) -{ 1 }-> 2 :|: z = 1 a__isNatList(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__length(z) -{ 2 }-> a__U11(a__and(a__and(a__isNat(V13), 1 + V2'), 1 + N), 1 + V13 + V2') :|: V13 >= 0, z = 1 + N + (1 + V13 + V2'), V2' >= 0, N >= 0 a__length(z) -{ 2 }-> a__U11(a__and(2, 1 + N), 1) :|: z = 1 + N + 1, N >= 0 a__length(z) -{ 2 }-> a__U11(a__and(1 + L, 1 + N), L) :|: z = 1 + N + L, L >= 0, N >= 0 a__length(z) -{ 1 }-> 0 :|: z = 1 a__length(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__zeros -{ 1 }-> 3 :|: a__zeros -{ 1 }-> 1 + 0 + 3 :|: mark(z) -{ 1 }-> a__zeros :|: z = 3 mark(z) -{ 2 }-> a__length(a__zeros) :|: z = 1 + 3 mark(z) -{ 2 }-> a__length(a__length(mark(X16))) :|: X16 >= 0, z = 1 + (1 + X16) mark(z) -{ 2 }-> a__length(a__isNatList(X19)) :|: X19 >= 0, z = 1 + (1 + X19) mark(z) -{ 2 }-> a__length(a__isNatIList(X20)) :|: z = 1 + (1 + X20), X20 >= 0 mark(z) -{ 2 }-> a__length(a__isNat(X18)) :|: X18 >= 0, z = 1 + (1 + X18) mark(z) -{ 2 }-> a__length(a__and(mark(X17), X26)) :|: z = 1 + (1 + X17 + X26), X26 >= 0, X17 >= 0 mark(z) -{ 2 }-> a__length(a__U11(mark(X15), X25)) :|: z = 1 + (1 + X15 + X25), X15 >= 0, X25 >= 0 mark(z) -{ 2 }-> a__length(2) :|: z = 1 + 2 mark(z) -{ 2 }-> a__length(1) :|: z = 1 + 1 mark(z) -{ 2 }-> a__length(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__length(1 + mark(X28)) :|: X28 >= 0, z = 1 + (1 + X28) mark(z) -{ 2 }-> a__length(1 + mark(X110) + X27) :|: X110 >= 0, X27 >= 0, z = 1 + (1 + X110 + X27) mark(z) -{ 1 }-> a__isNatList(X) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__isNatIList(X) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__isNat(X) :|: z = 1 + X, X >= 0 mark(z) -{ 2 }-> a__and(a__zeros, X2) :|: X2 >= 0, z = 1 + 3 + X2 mark(z) -{ 2 }-> a__and(a__length(mark(X30)), X2) :|: z = 1 + (1 + X30) + X2, X30 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__and(a__isNatList(X32), X2) :|: z = 1 + (1 + X32) + X2, X32 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__and(a__isNatIList(X33), X2) :|: X33 >= 0, z = 1 + (1 + X33) + X2, X2 >= 0 mark(z) -{ 2 }-> a__and(a__isNat(X31), X2) :|: z = 1 + (1 + X31) + X2, X31 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__and(a__and(mark(X112), X210), X2) :|: X112 >= 0, z = 1 + (1 + X112 + X210) + X2, X2 >= 0, X210 >= 0 mark(z) -{ 2 }-> a__and(a__U11(mark(X111), X29), X2) :|: X29 >= 0, X2 >= 0, z = 1 + (1 + X111 + X29) + X2, X111 >= 0 mark(z) -{ 2 }-> a__and(2, X2) :|: z = 1 + 2 + X2, X2 >= 0 mark(z) -{ 2 }-> a__and(1, X2) :|: z = 1 + 1 + X2, X2 >= 0 mark(z) -{ 2 }-> a__and(0, X2) :|: z = 1 + 0 + X2, X2 >= 0 mark(z) -{ 2 }-> a__and(1 + mark(X34), X2) :|: z = 1 + (1 + X34) + X2, X2 >= 0, X34 >= 0 mark(z) -{ 2 }-> a__and(1 + mark(X113) + X211, X2) :|: z = 1 + (1 + X113 + X211) + X2, X113 >= 0, X211 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__zeros, X2) :|: X2 >= 0, z = 1 + 3 + X2 mark(z) -{ 2 }-> a__U11(a__length(mark(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__isNatList(X8), X2) :|: z = 1 + (1 + X8) + X2, X8 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__isNatIList(X9), X2) :|: X9 >= 0, z = 1 + (1 + X9) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__isNat(X7), X2) :|: z = 1 + (1 + X7) + X2, X7 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__and(mark(X13), X23), X2) :|: z = 1 + (1 + X13 + X23) + X2, X2 >= 0, X13 >= 0, X23 >= 0 mark(z) -{ 2 }-> a__U11(a__U11(mark(X12), X22), X2) :|: z = 1 + (1 + X12 + X22) + X2, X12 >= 0, X22 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__U11(2, X2) :|: z = 1 + 2 + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(1, X2) :|: z = 1 + 1 + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(0, X2) :|: z = 1 + 0 + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(1 + mark(X10), X2) :|: X10 >= 0, z = 1 + (1 + X10) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(1 + mark(X14) + X24, X2) :|: z = 1 + (1 + X14 + X24) + X2, X2 >= 0, X14 >= 0, X24 >= 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 mark(z) -{ 1 }-> 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: a__zeros -{ 1 }-> 3 :|: a__zeros -{ 1 }-> 1 + 0 + 3 :|: ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: a__U11(z, z') -{ 2 }-> 1 + a__length(a__length(mark(X'))) :|: z = 2, X' >= 0, z' = 1 + X' a__U11(z, z') -{ 2 }-> 1 + a__length(a__isNatList(X3)) :|: z = 2, X3 >= 0, z' = 1 + X3 a__U11(z, z') -{ 2 }-> 1 + a__length(a__isNatIList(X4)) :|: z = 2, z' = 1 + X4, X4 >= 0 a__U11(z, z') -{ 2 }-> 1 + a__length(a__isNat(X'')) :|: z = 2, z' = 1 + X'', X'' >= 0 a__U11(z, z') -{ 2 }-> 1 + a__length(a__and(mark(X1''), X2'')) :|: z = 2, z' = 1 + X1'' + X2'', X1'' >= 0, X2'' >= 0 a__U11(z, z') -{ 2 }-> 1 + a__length(a__U11(mark(X1'), X2')) :|: z = 2, z' = 1 + X1' + X2', X2' >= 0, X1' >= 0 a__U11(z, z') -{ 3 }-> 1 + a__length(3) :|: z = 2, z' = 3 a__U11(z, z') -{ 2 }-> 1 + a__length(2) :|: z = 2, z' = 2 a__U11(z, z') -{ 2 }-> 1 + a__length(1) :|: z = 2, z' = 1 a__U11(z, z') -{ 2 }-> 1 + a__length(0) :|: z = 2, z' = 0 a__U11(z, z') -{ 2 }-> 1 + a__length(1 + mark(X5)) :|: z = 2, X5 >= 0, z' = 1 + X5 a__U11(z, z') -{ 2 }-> 1 + a__length(1 + mark(X11) + X21) :|: z = 2, z' = 1 + X11 + X21, X11 >= 0, X21 >= 0 a__U11(z, z') -{ 3 }-> 1 + a__length(1 + 0 + 3) :|: z = 2, z' = 3 a__U11(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__and(z, z') -{ 1 }-> mark(X) :|: z = 2, z' = X, X >= 0 a__and(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__isNat(z) -{ 1 }-> a__isNatList(V1) :|: z = 1 + V1, V1 >= 0 a__isNat(z) -{ 1 }-> a__isNat(V1) :|: z = 1 + V1, V1 >= 0 a__isNat(z) -{ 1 }-> 2 :|: z = 0 a__isNat(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__isNatIList(z) -{ 1 }-> a__isNatList(V) :|: z = V, V >= 0 a__isNatIList(z) -{ 2 }-> a__and(a__isNatList(V1'), 1 + V2) :|: V2 >= 0, z = 1 + (1 + V1') + V2, V1' >= 0 a__isNatIList(z) -{ 2 }-> a__and(a__isNat(V1''), 1 + V2) :|: V1'' >= 0, z = 1 + (1 + V1'') + V2, V2 >= 0 a__isNatIList(z) -{ 2 }-> a__and(2, 1 + V2) :|: V2 >= 0, z = 1 + 0 + V2 a__isNatIList(z) -{ 2 }-> a__and(1 + V1, 1 + V2) :|: V1 >= 0, V2 >= 0, z = 1 + V1 + V2 a__isNatIList(z) -{ 1 }-> 2 :|: z = 3 a__isNatIList(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__isNatList(z) -{ 2 }-> a__and(a__isNatList(V11), 1 + V2) :|: z = 1 + (1 + V11) + V2, V11 >= 0, V2 >= 0 a__isNatList(z) -{ 2 }-> a__and(a__isNat(V12), 1 + V2) :|: V12 >= 0, V2 >= 0, z = 1 + (1 + V12) + V2 a__isNatList(z) -{ 2 }-> a__and(2, 1 + V2) :|: V2 >= 0, z = 1 + 0 + V2 a__isNatList(z) -{ 2 }-> a__and(1 + V1, 1 + V2) :|: V1 >= 0, V2 >= 0, z = 1 + V1 + V2 a__isNatList(z) -{ 1 }-> 2 :|: z = 1 a__isNatList(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__length(z) -{ 2 }-> a__U11(a__and(a__and(a__isNat(V13), 1 + V2'), 1 + N), 1 + V13 + V2') :|: V13 >= 0, z = 1 + N + (1 + V13 + V2'), V2' >= 0, N >= 0 a__length(z) -{ 2 }-> a__U11(a__and(2, 1 + N), 1) :|: z = 1 + N + 1, N >= 0 a__length(z) -{ 2 }-> a__U11(a__and(1 + L, 1 + N), L) :|: z = 1 + N + L, L >= 0, N >= 0 a__length(z) -{ 1 }-> 0 :|: z = 1 a__length(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__zeros -{ 1 }-> 3 :|: a__zeros -{ 1 }-> 1 + 0 + 3 :|: mark(z) -{ 2 }-> a__length(a__length(mark(X16))) :|: X16 >= 0, z = 1 + (1 + X16) mark(z) -{ 2 }-> a__length(a__isNatList(X19)) :|: X19 >= 0, z = 1 + (1 + X19) mark(z) -{ 2 }-> a__length(a__isNatIList(X20)) :|: z = 1 + (1 + X20), X20 >= 0 mark(z) -{ 2 }-> a__length(a__isNat(X18)) :|: X18 >= 0, z = 1 + (1 + X18) mark(z) -{ 2 }-> a__length(a__and(mark(X17), X26)) :|: z = 1 + (1 + X17 + X26), X26 >= 0, X17 >= 0 mark(z) -{ 2 }-> a__length(a__U11(mark(X15), X25)) :|: z = 1 + (1 + X15 + X25), X15 >= 0, X25 >= 0 mark(z) -{ 3 }-> a__length(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__length(2) :|: z = 1 + 2 mark(z) -{ 2 }-> a__length(1) :|: z = 1 + 1 mark(z) -{ 2 }-> a__length(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__length(1 + mark(X28)) :|: X28 >= 0, z = 1 + (1 + X28) mark(z) -{ 2 }-> a__length(1 + mark(X110) + X27) :|: X110 >= 0, X27 >= 0, z = 1 + (1 + X110 + X27) mark(z) -{ 3 }-> a__length(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 1 }-> a__isNatList(X) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__isNatIList(X) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__isNat(X) :|: z = 1 + X, X >= 0 mark(z) -{ 2 }-> a__and(a__length(mark(X30)), X2) :|: z = 1 + (1 + X30) + X2, X30 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__and(a__isNatList(X32), X2) :|: z = 1 + (1 + X32) + X2, X32 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__and(a__isNatIList(X33), X2) :|: X33 >= 0, z = 1 + (1 + X33) + X2, X2 >= 0 mark(z) -{ 2 }-> a__and(a__isNat(X31), X2) :|: z = 1 + (1 + X31) + X2, X31 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__and(a__and(mark(X112), X210), X2) :|: X112 >= 0, z = 1 + (1 + X112 + X210) + X2, X2 >= 0, X210 >= 0 mark(z) -{ 2 }-> a__and(a__U11(mark(X111), X29), X2) :|: X29 >= 0, X2 >= 0, z = 1 + (1 + X111 + X29) + X2, X111 >= 0 mark(z) -{ 3 }-> a__and(3, X2) :|: X2 >= 0, z = 1 + 3 + X2 mark(z) -{ 2 }-> a__and(2, X2) :|: z = 1 + 2 + X2, X2 >= 0 mark(z) -{ 2 }-> a__and(1, X2) :|: z = 1 + 1 + X2, X2 >= 0 mark(z) -{ 2 }-> a__and(0, X2) :|: z = 1 + 0 + X2, X2 >= 0 mark(z) -{ 2 }-> a__and(1 + mark(X34), X2) :|: z = 1 + (1 + X34) + X2, X2 >= 0, X34 >= 0 mark(z) -{ 2 }-> a__and(1 + mark(X113) + X211, X2) :|: z = 1 + (1 + X113 + X211) + X2, X113 >= 0, X211 >= 0, X2 >= 0 mark(z) -{ 3 }-> a__and(1 + 0 + 3, X2) :|: X2 >= 0, z = 1 + 3 + X2 mark(z) -{ 2 }-> a__U11(a__length(mark(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__isNatList(X8), X2) :|: z = 1 + (1 + X8) + X2, X8 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__isNatIList(X9), X2) :|: X9 >= 0, z = 1 + (1 + X9) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__isNat(X7), X2) :|: z = 1 + (1 + X7) + X2, X7 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__and(mark(X13), X23), X2) :|: z = 1 + (1 + X13 + X23) + X2, X2 >= 0, X13 >= 0, X23 >= 0 mark(z) -{ 2 }-> a__U11(a__U11(mark(X12), X22), X2) :|: z = 1 + (1 + X12 + X22) + X2, X12 >= 0, X22 >= 0, X2 >= 0 mark(z) -{ 3 }-> a__U11(3, X2) :|: X2 >= 0, z = 1 + 3 + X2 mark(z) -{ 2 }-> a__U11(2, X2) :|: z = 1 + 2 + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(1, X2) :|: z = 1 + 1 + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(0, X2) :|: z = 1 + 0 + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(1 + mark(X10), X2) :|: X10 >= 0, z = 1 + (1 + X10) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(1 + mark(X14) + X24, X2) :|: z = 1 + (1 + X14 + X24) + X2, X2 >= 0, X14 >= 0, X24 >= 0 mark(z) -{ 3 }-> a__U11(1 + 0 + 3, X2) :|: X2 >= 0, z = 1 + 3 + X2 mark(z) -{ 2 }-> 3 :|: z = 3 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 mark(z) -{ 1 }-> 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 2 }-> 1 + 0 + 3 :|: z = 3 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: a__U11(z, z') -{ 2 }-> 1 + a__length(a__length(mark(z' - 1))) :|: z = 2, z' - 1 >= 0 a__U11(z, z') -{ 2 }-> 1 + a__length(a__isNatList(z' - 1)) :|: z = 2, z' - 1 >= 0 a__U11(z, z') -{ 2 }-> 1 + a__length(a__isNatIList(z' - 1)) :|: z = 2, z' - 1 >= 0 a__U11(z, z') -{ 2 }-> 1 + a__length(a__isNat(z' - 1)) :|: z = 2, z' - 1 >= 0 a__U11(z, z') -{ 2 }-> 1 + a__length(a__and(mark(X1''), X2'')) :|: z = 2, z' = 1 + X1'' + X2'', X1'' >= 0, X2'' >= 0 a__U11(z, z') -{ 2 }-> 1 + a__length(a__U11(mark(X1'), X2')) :|: z = 2, z' = 1 + X1' + X2', X2' >= 0, X1' >= 0 a__U11(z, z') -{ 3 }-> 1 + a__length(3) :|: z = 2, z' = 3 a__U11(z, z') -{ 2 }-> 1 + a__length(2) :|: z = 2, z' = 2 a__U11(z, z') -{ 2 }-> 1 + a__length(1) :|: z = 2, z' = 1 a__U11(z, z') -{ 2 }-> 1 + a__length(0) :|: z = 2, z' = 0 a__U11(z, z') -{ 2 }-> 1 + a__length(1 + mark(z' - 1)) :|: z = 2, z' - 1 >= 0 a__U11(z, z') -{ 2 }-> 1 + a__length(1 + mark(X11) + X21) :|: z = 2, z' = 1 + X11 + X21, X11 >= 0, X21 >= 0 a__U11(z, z') -{ 3 }-> 1 + a__length(1 + 0 + 3) :|: z = 2, z' = 3 a__U11(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 a__and(z, z') -{ 1 }-> mark(z') :|: z = 2, z' >= 0 a__and(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 a__isNat(z) -{ 1 }-> a__isNatList(z - 1) :|: z - 1 >= 0 a__isNat(z) -{ 1 }-> a__isNat(z - 1) :|: z - 1 >= 0 a__isNat(z) -{ 1 }-> 2 :|: z = 0 a__isNat(z) -{ 1 }-> 1 + z :|: z >= 0 a__isNatIList(z) -{ 1 }-> a__isNatList(z) :|: z >= 0 a__isNatIList(z) -{ 2 }-> a__and(a__isNatList(V1'), 1 + V2) :|: V2 >= 0, z = 1 + (1 + V1') + V2, V1' >= 0 a__isNatIList(z) -{ 2 }-> a__and(a__isNat(V1''), 1 + V2) :|: V1'' >= 0, z = 1 + (1 + V1'') + V2, V2 >= 0 a__isNatIList(z) -{ 2 }-> a__and(2, 1 + (z - 1)) :|: z - 1 >= 0 a__isNatIList(z) -{ 2 }-> a__and(1 + V1, 1 + V2) :|: V1 >= 0, V2 >= 0, z = 1 + V1 + V2 a__isNatIList(z) -{ 1 }-> 2 :|: z = 3 a__isNatIList(z) -{ 1 }-> 1 + z :|: z >= 0 a__isNatList(z) -{ 2 }-> a__and(a__isNatList(V11), 1 + V2) :|: z = 1 + (1 + V11) + V2, V11 >= 0, V2 >= 0 a__isNatList(z) -{ 2 }-> a__and(a__isNat(V12), 1 + V2) :|: V12 >= 0, V2 >= 0, z = 1 + (1 + V12) + V2 a__isNatList(z) -{ 2 }-> a__and(2, 1 + (z - 1)) :|: z - 1 >= 0 a__isNatList(z) -{ 2 }-> a__and(1 + V1, 1 + V2) :|: V1 >= 0, V2 >= 0, z = 1 + V1 + V2 a__isNatList(z) -{ 1 }-> 2 :|: z = 1 a__isNatList(z) -{ 1 }-> 1 + z :|: z >= 0 a__length(z) -{ 2 }-> a__U11(a__and(a__and(a__isNat(V13), 1 + V2'), 1 + N), 1 + V13 + V2') :|: V13 >= 0, z = 1 + N + (1 + V13 + V2'), V2' >= 0, N >= 0 a__length(z) -{ 2 }-> a__U11(a__and(2, 1 + (z - 2)), 1) :|: z - 2 >= 0 a__length(z) -{ 2 }-> a__U11(a__and(1 + L, 1 + N), L) :|: z = 1 + N + L, L >= 0, N >= 0 a__length(z) -{ 1 }-> 0 :|: z = 1 a__length(z) -{ 1 }-> 1 + z :|: z >= 0 a__zeros -{ 1 }-> 3 :|: a__zeros -{ 1 }-> 1 + 0 + 3 :|: mark(z) -{ 2 }-> a__length(a__length(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__length(a__isNatList(z - 2)) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__length(a__isNatIList(z - 2)) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__length(a__isNat(z - 2)) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__length(a__and(mark(X17), X26)) :|: z = 1 + (1 + X17 + X26), X26 >= 0, X17 >= 0 mark(z) -{ 2 }-> a__length(a__U11(mark(X15), X25)) :|: z = 1 + (1 + X15 + X25), X15 >= 0, X25 >= 0 mark(z) -{ 3 }-> a__length(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__length(2) :|: z = 1 + 2 mark(z) -{ 2 }-> a__length(1) :|: z = 1 + 1 mark(z) -{ 2 }-> a__length(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__length(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__length(1 + mark(X110) + X27) :|: X110 >= 0, X27 >= 0, z = 1 + (1 + X110 + X27) mark(z) -{ 3 }-> a__length(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 1 }-> a__isNatList(z - 1) :|: z - 1 >= 0 mark(z) -{ 1 }-> a__isNatIList(z - 1) :|: z - 1 >= 0 mark(z) -{ 1 }-> a__isNat(z - 1) :|: z - 1 >= 0 mark(z) -{ 2 }-> a__and(a__length(mark(X30)), X2) :|: z = 1 + (1 + X30) + X2, X30 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__and(a__isNatList(X32), X2) :|: z = 1 + (1 + X32) + X2, X32 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__and(a__isNatIList(X33), X2) :|: X33 >= 0, z = 1 + (1 + X33) + X2, X2 >= 0 mark(z) -{ 2 }-> a__and(a__isNat(X31), X2) :|: z = 1 + (1 + X31) + X2, X31 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__and(a__and(mark(X112), X210), X2) :|: X112 >= 0, z = 1 + (1 + X112 + X210) + X2, X2 >= 0, X210 >= 0 mark(z) -{ 2 }-> a__and(a__U11(mark(X111), X29), X2) :|: X29 >= 0, X2 >= 0, z = 1 + (1 + X111 + X29) + X2, X111 >= 0 mark(z) -{ 3 }-> a__and(3, z - 4) :|: z - 4 >= 0 mark(z) -{ 2 }-> a__and(2, z - 3) :|: z - 3 >= 0 mark(z) -{ 2 }-> a__and(1, z - 2) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__and(0, z - 1) :|: z - 1 >= 0 mark(z) -{ 2 }-> a__and(1 + mark(X34), X2) :|: z = 1 + (1 + X34) + X2, X2 >= 0, X34 >= 0 mark(z) -{ 2 }-> a__and(1 + mark(X113) + X211, X2) :|: z = 1 + (1 + X113 + X211) + X2, X113 >= 0, X211 >= 0, X2 >= 0 mark(z) -{ 3 }-> a__and(1 + 0 + 3, z - 4) :|: z - 4 >= 0 mark(z) -{ 2 }-> a__U11(a__length(mark(X6)), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__isNatList(X8), X2) :|: z = 1 + (1 + X8) + X2, X8 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__isNatIList(X9), X2) :|: X9 >= 0, z = 1 + (1 + X9) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__isNat(X7), X2) :|: z = 1 + (1 + X7) + X2, X7 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__and(mark(X13), X23), X2) :|: z = 1 + (1 + X13 + X23) + X2, X2 >= 0, X13 >= 0, X23 >= 0 mark(z) -{ 2 }-> a__U11(a__U11(mark(X12), X22), X2) :|: z = 1 + (1 + X12 + X22) + X2, X12 >= 0, X22 >= 0, X2 >= 0 mark(z) -{ 3 }-> a__U11(3, z - 4) :|: z - 4 >= 0 mark(z) -{ 2 }-> a__U11(2, z - 3) :|: z - 3 >= 0 mark(z) -{ 2 }-> a__U11(1, z - 2) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__U11(0, z - 1) :|: z - 1 >= 0 mark(z) -{ 2 }-> a__U11(1 + mark(X10), X2) :|: X10 >= 0, z = 1 + (1 + X10) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(1 + mark(X14) + X24, X2) :|: z = 1 + (1 + X14 + X24) + X2, X2 >= 0, X14 >= 0, X24 >= 0 mark(z) -{ 3 }-> a__U11(1 + 0 + 3, z - 4) :|: z - 4 >= 0 mark(z) -{ 2 }-> 3 :|: z = 3 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 mark(z) -{ 1 }-> 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 2 }-> 1 + 0 + 3 :|: z = 3 ---------------------------------------- (19) 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 ---------------------------------------- (20) 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__zeros -> cons(0, zeros) [1] a__U11(tt, L) -> s(a__length(mark(L))) [1] a__and(tt, X) -> mark(X) [1] a__isNat(0) -> tt [1] a__isNat(length(V1)) -> a__isNatList(V1) [1] a__isNat(s(V1)) -> a__isNat(V1) [1] a__isNatIList(V) -> a__isNatList(V) [1] a__isNatIList(zeros) -> tt [1] a__isNatIList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatIList(V2)) [1] a__isNatList(nil) -> tt [1] a__isNatList(cons(V1, V2)) -> a__and(a__isNat(V1), isNatList(V2)) [1] a__length(nil) -> 0 [1] a__length(cons(N, L)) -> a__U11(a__and(a__isNatList(L), isNat(N)), L) [1] mark(zeros) -> a__zeros [1] mark(U11(X1, X2)) -> a__U11(mark(X1), X2) [1] mark(length(X)) -> a__length(mark(X)) [1] mark(and(X1, X2)) -> a__and(mark(X1), X2) [1] mark(isNat(X)) -> a__isNat(X) [1] mark(isNatList(X)) -> a__isNatList(X) [1] mark(isNatIList(X)) -> a__isNatIList(X) [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(0) -> 0 [1] mark(tt) -> tt [1] mark(s(X)) -> s(mark(X)) [1] mark(nil) -> nil [1] a__zeros -> zeros [1] a__U11(X1, X2) -> U11(X1, X2) [1] a__length(X) -> length(X) [1] a__and(X1, X2) -> and(X1, X2) [1] a__isNat(X) -> isNat(X) [1] a__isNatList(X) -> isNatList(X) [1] a__isNatIList(X) -> isNatIList(X) [1] The TRS has the following type information: a__zeros :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and cons :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and 0 :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and zeros :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__U11 :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and tt :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and s :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__length :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and mark :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__and :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__isNat :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and length :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__isNatList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and a__isNatIList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and isNatIList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and nil :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and isNatList :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and isNat :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and U11 :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and and :: 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and -> 0:zeros:cons:tt:s:length:isNatIList:nil:isNatList:isNat:U11:and Rewrite Strategy: INNERMOST ---------------------------------------- (21) 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 zeros => 3 tt => 2 nil => 1 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: a__U11(z, z') -{ 1 }-> 1 + a__length(mark(L)) :|: z = 2, L >= 0, z' = L a__U11(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__and(z, z') -{ 1 }-> mark(X) :|: z = 2, z' = X, X >= 0 a__and(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__isNat(z) -{ 1 }-> a__isNatList(V1) :|: z = 1 + V1, V1 >= 0 a__isNat(z) -{ 1 }-> a__isNat(V1) :|: z = 1 + V1, V1 >= 0 a__isNat(z) -{ 1 }-> 2 :|: z = 0 a__isNat(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__isNatIList(z) -{ 1 }-> a__isNatList(V) :|: z = V, V >= 0 a__isNatIList(z) -{ 1 }-> a__and(a__isNat(V1), 1 + V2) :|: V1 >= 0, V2 >= 0, z = 1 + V1 + V2 a__isNatIList(z) -{ 1 }-> 2 :|: z = 3 a__isNatIList(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__isNatList(z) -{ 1 }-> a__and(a__isNat(V1), 1 + V2) :|: V1 >= 0, V2 >= 0, z = 1 + V1 + V2 a__isNatList(z) -{ 1 }-> 2 :|: z = 1 a__isNatList(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__length(z) -{ 1 }-> a__U11(a__and(a__isNatList(L), 1 + N), L) :|: z = 1 + N + L, L >= 0, N >= 0 a__length(z) -{ 1 }-> 0 :|: z = 1 a__length(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__zeros -{ 1 }-> 3 :|: a__zeros -{ 1 }-> 1 + 0 + 3 :|: mark(z) -{ 1 }-> a__zeros :|: z = 3 mark(z) -{ 1 }-> a__length(mark(X)) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__isNatList(X) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__isNatIList(X) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__isNat(X) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__and(mark(X1), X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 1 }-> a__U11(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 mark(z) -{ 1 }-> 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (23) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__ZEROS -> c A__ZEROS -> c1 A__U11(tt, z0) -> c2(A__LENGTH(mark(z0)), MARK(z0)) A__U11(z0, z1) -> c3 A__AND(tt, z0) -> c4(MARK(z0)) A__AND(z0, z1) -> c5 A__ISNAT(0) -> c6 A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNAT(z0) -> c9 A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__ISNATILIST(zeros) -> c11 A__ISNATILIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), isNatIList(z1)), A__ISNAT(z0)) A__ISNATILIST(z0) -> c13 A__ISNATLIST(nil) -> c14 A__ISNATLIST(cons(z0, z1)) -> c15(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0)) A__ISNATLIST(z0) -> c16 A__LENGTH(nil) -> c17 A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) A__LENGTH(z0) -> c19 MARK(zeros) -> c20(A__ZEROS) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(0) -> c28 MARK(tt) -> c29 MARK(s(z0)) -> c30(MARK(z0)) MARK(nil) -> c31 S tuples: A__ZEROS -> c A__ZEROS -> c1 A__U11(tt, z0) -> c2(A__LENGTH(mark(z0)), MARK(z0)) A__U11(z0, z1) -> c3 A__AND(tt, z0) -> c4(MARK(z0)) A__AND(z0, z1) -> c5 A__ISNAT(0) -> c6 A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNAT(z0) -> c9 A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__ISNATILIST(zeros) -> c11 A__ISNATILIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), isNatIList(z1)), A__ISNAT(z0)) A__ISNATILIST(z0) -> c13 A__ISNATLIST(nil) -> c14 A__ISNATLIST(cons(z0, z1)) -> c15(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0)) A__ISNATLIST(z0) -> c16 A__LENGTH(nil) -> c17 A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) A__LENGTH(z0) -> c19 MARK(zeros) -> c20(A__ZEROS) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(0) -> c28 MARK(tt) -> c29 MARK(s(z0)) -> c30(MARK(z0)) MARK(nil) -> c31 K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__ZEROS, A__U11_2, A__AND_2, A__ISNAT_1, A__ISNATILIST_1, A__ISNATLIST_1, A__LENGTH_1, MARK_1 Compound Symbols: c, c1, c2_2, c3, c4_1, c5, c6, c7_1, c8_1, c9, c10_1, c11, c12_2, c13, c14, c15_2, c16, c17, c18_3, c19, c20_1, c21_2, c22_2, c23_2, c24_1, c25_1, c26_1, c27_1, c28, c29, c30_1, c31 ---------------------------------------- (25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 16 trailing nodes: MARK(0) -> c28 MARK(nil) -> c31 A__U11(z0, z1) -> c3 A__ISNATLIST(z0) -> c16 A__ZEROS -> c A__LENGTH(nil) -> c17 A__ISNATILIST(zeros) -> c11 A__ISNAT(0) -> c6 A__AND(z0, z1) -> c5 A__ZEROS -> c1 A__ISNATILIST(z0) -> c13 MARK(tt) -> c29 A__ISNAT(z0) -> c9 MARK(zeros) -> c20(A__ZEROS) A__ISNATLIST(nil) -> c14 A__LENGTH(z0) -> c19 ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__U11(tt, z0) -> c2(A__LENGTH(mark(z0)), MARK(z0)) A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__ISNATILIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), isNatIList(z1)), A__ISNAT(z0)) A__ISNATLIST(cons(z0, z1)) -> c15(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) S tuples: A__U11(tt, z0) -> c2(A__LENGTH(mark(z0)), MARK(z0)) A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__ISNATILIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), isNatIList(z1)), A__ISNAT(z0)) A__ISNATLIST(cons(z0, z1)) -> c15(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__AND_2, A__ISNAT_1, A__ISNATILIST_1, A__ISNATLIST_1, A__LENGTH_1, MARK_1 Compound Symbols: c2_2, c4_1, c7_1, c8_1, c10_1, c12_2, c15_2, c18_3, c21_2, c22_2, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1 ---------------------------------------- (27) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U11(tt, z0) -> c2(A__LENGTH(mark(z0)), MARK(z0)) by A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros), MARK(zeros)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, 0) -> c2(A__LENGTH(0), MARK(0)) A__U11(tt, tt) -> c2(A__LENGTH(tt), MARK(tt)) A__U11(tt, s(z0)) -> c2(A__LENGTH(s(mark(z0))), MARK(s(z0))) A__U11(tt, nil) -> c2(A__LENGTH(nil), MARK(nil)) ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__ISNATILIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), isNatIList(z1)), A__ISNAT(z0)) A__ISNATLIST(cons(z0, z1)) -> c15(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros), MARK(zeros)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, 0) -> c2(A__LENGTH(0), MARK(0)) A__U11(tt, tt) -> c2(A__LENGTH(tt), MARK(tt)) A__U11(tt, s(z0)) -> c2(A__LENGTH(s(mark(z0))), MARK(s(z0))) A__U11(tt, nil) -> c2(A__LENGTH(nil), MARK(nil)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__ISNATILIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), isNatIList(z1)), A__ISNAT(z0)) A__ISNATLIST(cons(z0, z1)) -> c15(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros), MARK(zeros)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, 0) -> c2(A__LENGTH(0), MARK(0)) A__U11(tt, tt) -> c2(A__LENGTH(tt), MARK(tt)) A__U11(tt, s(z0)) -> c2(A__LENGTH(s(mark(z0))), MARK(s(z0))) A__U11(tt, nil) -> c2(A__LENGTH(nil), MARK(nil)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, A__ISNATLIST_1, A__LENGTH_1, MARK_1, A__U11_2 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c12_2, c15_2, c18_3, c21_2, c22_2, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2 ---------------------------------------- (29) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: A__U11(tt, 0) -> c2(A__LENGTH(0), MARK(0)) A__U11(tt, tt) -> c2(A__LENGTH(tt), MARK(tt)) A__U11(tt, nil) -> c2(A__LENGTH(nil), MARK(nil)) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__ISNATILIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), isNatIList(z1)), A__ISNAT(z0)) A__ISNATLIST(cons(z0, z1)) -> c15(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros), MARK(zeros)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(A__LENGTH(s(mark(z0))), MARK(s(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__ISNATILIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), isNatIList(z1)), A__ISNAT(z0)) A__ISNATLIST(cons(z0, z1)) -> c15(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros), MARK(zeros)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(A__LENGTH(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, A__ISNATLIST_1, A__LENGTH_1, MARK_1, A__U11_2 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c12_2, c15_2, c18_3, c21_2, c22_2, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2 ---------------------------------------- (31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__ISNATILIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), isNatIList(z1)), A__ISNAT(z0)) A__ISNATLIST(cons(z0, z1)) -> c15(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__ISNATILIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), isNatIList(z1)), A__ISNAT(z0)) A__ISNATLIST(cons(z0, z1)) -> c15(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, A__ISNATLIST_1, A__LENGTH_1, MARK_1, A__U11_2 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c12_2, c15_2, c18_3, c21_2, c22_2, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1 ---------------------------------------- (33) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__ISNATILIST(cons(z0, z1)) -> c12(A__AND(a__isNat(z0), isNatIList(z1)), A__ISNAT(z0)) by A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(0)) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(z0, x1)) -> c12(A__AND(isNat(z0), isNatIList(x1)), A__ISNAT(z0)) ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__ISNATLIST(cons(z0, z1)) -> c15(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(0)) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(z0, x1)) -> c12(A__AND(isNat(z0), isNatIList(x1)), A__ISNAT(z0)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__ISNATLIST(cons(z0, z1)) -> c15(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(0)) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(z0, x1)) -> c12(A__AND(isNat(z0), isNatIList(x1)), A__ISNAT(z0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, A__ISNATLIST_1, A__LENGTH_1, MARK_1, A__U11_2 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c15_2, c18_3, c21_2, c22_2, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2 ---------------------------------------- (35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__ISNATLIST(cons(z0, z1)) -> c15(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__ISNATLIST(cons(z0, z1)) -> c15(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, A__ISNATLIST_1, A__LENGTH_1, MARK_1, A__U11_2 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c15_2, c18_3, c21_2, c22_2, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1 ---------------------------------------- (37) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__ISNATLIST(cons(z0, z1)) -> c15(A__AND(a__isNat(z0), isNatList(z1)), A__ISNAT(z0)) by A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(z0, x1)) -> c15(A__AND(isNat(z0), isNatList(x1)), A__ISNAT(z0)) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(z0, x1)) -> c15(A__AND(isNat(z0), isNatList(x1)), A__ISNAT(z0)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(z0, x1)) -> c15(A__AND(isNat(z0), isNatList(x1)), A__ISNAT(z0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, A__LENGTH_1, MARK_1, A__U11_2, A__ISNATLIST_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c18_3, c21_2, c22_2, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, A__LENGTH_1, MARK_1, A__U11_2, A__ISNATLIST_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c18_3, c21_2, c22_2, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1 ---------------------------------------- (41) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__LENGTH(cons(z0, z1)) -> c18(A__U11(a__and(a__isNatList(z1), isNat(z0)), z1), A__AND(a__isNatList(z1), isNat(z0)), A__ISNATLIST(z1)) by A__LENGTH(cons(x0, x1)) -> c18(A__U11(and(a__isNatList(x1), isNat(x0)), x1), A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__U11(a__and(tt, isNat(x0)), nil), A__AND(a__isNatList(nil), isNat(x0)), A__ISNATLIST(nil)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__U11(and(a__isNatList(x1), isNat(x0)), x1), A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__U11(a__and(tt, isNat(x0)), nil), A__AND(a__isNatList(nil), isNat(x0)), A__ISNATLIST(nil)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__U11(and(a__isNatList(x1), isNat(x0)), x1), A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__U11(a__and(tt, isNat(x0)), nil), A__AND(a__isNatList(nil), isNat(x0)), A__ISNATLIST(nil)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c21_2, c22_2, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3 ---------------------------------------- (43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c21_2, c22_2, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(z0, z1)) -> c21(A__U11(mark(z0), z1), MARK(z0)) by MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1), MARK(zeros)) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(cons(z0, z1), x1)) -> c21(A__U11(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(U11(0, x1)) -> c21(A__U11(0, x1), MARK(0)) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1), MARK(tt)) MARK(U11(s(z0), x1)) -> c21(A__U11(s(mark(z0)), x1), MARK(s(z0))) MARK(U11(nil, x1)) -> c21(A__U11(nil, x1), MARK(nil)) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1), MARK(zeros)) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(cons(z0, z1), x1)) -> c21(A__U11(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(U11(0, x1)) -> c21(A__U11(0, x1), MARK(0)) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1), MARK(tt)) MARK(U11(s(z0), x1)) -> c21(A__U11(s(mark(z0)), x1), MARK(s(z0))) MARK(U11(nil, x1)) -> c21(A__U11(nil, x1), MARK(nil)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1), MARK(zeros)) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(cons(z0, z1), x1)) -> c21(A__U11(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(U11(0, x1)) -> c21(A__U11(0, x1), MARK(0)) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1), MARK(tt)) MARK(U11(s(z0), x1)) -> c21(A__U11(s(mark(z0)), x1), MARK(s(z0))) MARK(U11(nil, x1)) -> c21(A__U11(nil, x1), MARK(nil)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c22_2, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2 ---------------------------------------- (47) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(U11(0, x1)) -> c21(A__U11(0, x1), MARK(0)) MARK(U11(nil, x1)) -> c21(A__U11(nil, x1), MARK(nil)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1), MARK(zeros)) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(cons(z0, z1), x1)) -> c21(A__U11(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1), MARK(tt)) MARK(U11(s(z0), x1)) -> c21(A__U11(s(mark(z0)), x1), MARK(s(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1), MARK(zeros)) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(cons(z0, z1), x1)) -> c21(A__U11(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1), MARK(tt)) MARK(U11(s(z0), x1)) -> c21(A__U11(s(mark(z0)), x1), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c22_2, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c22_2, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(z0)) -> c22(A__LENGTH(mark(z0)), MARK(z0)) by MARK(length(zeros)) -> c22(A__LENGTH(a__zeros), MARK(zeros)) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(0)) -> c22(A__LENGTH(0), MARK(0)) MARK(length(tt)) -> c22(A__LENGTH(tt), MARK(tt)) MARK(length(s(z0))) -> c22(A__LENGTH(s(mark(z0))), MARK(s(z0))) MARK(length(nil)) -> c22(A__LENGTH(nil), MARK(nil)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros), MARK(zeros)) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(0)) -> c22(A__LENGTH(0), MARK(0)) MARK(length(tt)) -> c22(A__LENGTH(tt), MARK(tt)) MARK(length(s(z0))) -> c22(A__LENGTH(s(mark(z0))), MARK(s(z0))) MARK(length(nil)) -> c22(A__LENGTH(nil), MARK(nil)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros), MARK(zeros)) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(0)) -> c22(A__LENGTH(0), MARK(0)) MARK(length(tt)) -> c22(A__LENGTH(tt), MARK(tt)) MARK(length(s(z0))) -> c22(A__LENGTH(s(mark(z0))), MARK(s(z0))) MARK(length(nil)) -> c22(A__LENGTH(nil), MARK(nil)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2 ---------------------------------------- (53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(length(nil)) -> c22(A__LENGTH(nil), MARK(nil)) MARK(length(0)) -> c22(A__LENGTH(0), MARK(0)) MARK(length(tt)) -> c22(A__LENGTH(tt), MARK(tt)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros), MARK(zeros)) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(A__LENGTH(s(mark(z0))), MARK(s(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros), MARK(zeros)) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(A__LENGTH(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2 ---------------------------------------- (55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c23_2, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(and(z0, z1)) -> c23(A__AND(mark(z0), z1), MARK(z0)) by MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1), MARK(zeros)) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(cons(z0, z1), x1)) -> c23(A__AND(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(and(0, x1)) -> c23(A__AND(0, x1), MARK(0)) MARK(and(tt, x1)) -> c23(A__AND(tt, x1), MARK(tt)) MARK(and(s(z0), x1)) -> c23(A__AND(s(mark(z0)), x1), MARK(s(z0))) MARK(and(nil, x1)) -> c23(A__AND(nil, x1), MARK(nil)) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1), MARK(zeros)) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(cons(z0, z1), x1)) -> c23(A__AND(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(and(0, x1)) -> c23(A__AND(0, x1), MARK(0)) MARK(and(tt, x1)) -> c23(A__AND(tt, x1), MARK(tt)) MARK(and(s(z0), x1)) -> c23(A__AND(s(mark(z0)), x1), MARK(s(z0))) MARK(and(nil, x1)) -> c23(A__AND(nil, x1), MARK(nil)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1), MARK(zeros)) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(cons(z0, z1), x1)) -> c23(A__AND(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(and(0, x1)) -> c23(A__AND(0, x1), MARK(0)) MARK(and(tt, x1)) -> c23(A__AND(tt, x1), MARK(tt)) MARK(and(s(z0), x1)) -> c23(A__AND(s(mark(z0)), x1), MARK(s(z0))) MARK(and(nil, x1)) -> c23(A__AND(nil, x1), MARK(nil)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2 ---------------------------------------- (59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(and(0, x1)) -> c23(A__AND(0, x1), MARK(0)) MARK(and(nil, x1)) -> c23(A__AND(nil, x1), MARK(nil)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1), MARK(zeros)) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(cons(z0, z1), x1)) -> c23(A__AND(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1), MARK(tt)) MARK(and(s(z0), x1)) -> c23(A__AND(s(mark(z0)), x1), MARK(s(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1), MARK(zeros)) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(cons(z0, z1), x1)) -> c23(A__AND(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1), MARK(tt)) MARK(and(s(z0), x1)) -> c23(A__AND(s(mark(z0)), x1), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U11(tt, U11(z0, z1)) -> c2(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) by A__U11(tt, U11(x0, z1)) -> c2(A__LENGTH(U11(mark(x0), z1)), MARK(U11(x0, z1))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1)), MARK(U11(0, x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1)), MARK(U11(nil, x1))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(x0, z1)) -> c2(A__LENGTH(U11(mark(x0), z1)), MARK(U11(x0, z1))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1)), MARK(U11(0, x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1)), MARK(U11(nil, x1))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(x0, z1)) -> c2(A__LENGTH(U11(mark(x0), z1)), MARK(U11(x0, z1))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1)), MARK(U11(0, x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1)), MARK(U11(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U11(tt, length(z0)) -> c2(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) by A__U11(tt, length(x0)) -> c2(A__LENGTH(length(mark(x0))), MARK(length(x0))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0)), MARK(length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt)), MARK(length(tt))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil)), MARK(length(nil))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(x0)) -> c2(A__LENGTH(length(mark(x0))), MARK(length(x0))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0)), MARK(length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt)), MARK(length(tt))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil)), MARK(length(nil))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(x0)) -> c2(A__LENGTH(length(mark(x0))), MARK(length(x0))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0)), MARK(length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt)), MARK(length(tt))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil)), MARK(length(nil))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U11(tt, and(z0, z1)) -> c2(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) by A__U11(tt, and(x0, z1)) -> c2(A__LENGTH(and(mark(x0), z1)), MARK(and(x0, z1))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1)), MARK(and(0, x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1)), MARK(and(nil, x1))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(x0, z1)) -> c2(A__LENGTH(and(mark(x0), z1)), MARK(and(x0, z1))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1)), MARK(and(0, x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1)), MARK(and(nil, x1))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(x0, z1)) -> c2(A__LENGTH(and(mark(x0), z1)), MARK(and(x0, z1))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1)), MARK(and(0, x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1)), MARK(and(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U11(tt, isNat(z0)) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) by A__U11(tt, isNat(0)) -> c2(A__LENGTH(tt), MARK(isNat(0))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(isNat(z0)), MARK(isNat(z0))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(0)) -> c2(A__LENGTH(tt), MARK(isNat(0))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(isNat(z0)), MARK(isNat(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(0)) -> c2(A__LENGTH(tt), MARK(isNat(0))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(z0)) -> c2(A__LENGTH(isNat(z0)), MARK(isNat(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) by A__U11(tt, isNatList(nil)) -> c2(A__LENGTH(tt), MARK(isNatList(nil))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(isNatList(z0)), MARK(isNatList(z0))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(nil)) -> c2(A__LENGTH(tt), MARK(isNatList(nil))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(isNatList(z0)), MARK(isNatList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(nil)) -> c2(A__LENGTH(tt), MARK(isNatList(nil))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(z0)) -> c2(A__LENGTH(isNatList(z0)), MARK(isNatList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) by A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(zeros)) -> c2(A__LENGTH(tt), MARK(isNatIList(zeros))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(isNatIList(z0)), MARK(isNatIList(z0))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(zeros)) -> c2(A__LENGTH(tt), MARK(isNatIList(zeros))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(isNatIList(z0)), MARK(isNatIList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(zeros)) -> c2(A__LENGTH(tt), MARK(isNatIList(zeros))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(isNatIList(z0)), MARK(isNatIList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (85) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U11(tt, zeros) -> c2(A__LENGTH(a__zeros)) by A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__U11(tt, zeros) -> c2(A__LENGTH(zeros)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__U11(tt, zeros) -> c2(A__LENGTH(zeros)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__U11(tt, zeros) -> c2(A__LENGTH(zeros)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U11(tt, zeros) -> c2(A__LENGTH(zeros)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) by A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__AND(isNatList(z0), isNatIList(x1)), A__ISNAT(length(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_2, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) by A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__AND(isNat(z0), isNatIList(x1)), A__ISNAT(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2 ---------------------------------------- (97) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) by A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__AND(isNatList(z0), isNatList(x1)), A__ISNAT(length(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2 ---------------------------------------- (101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_2, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) by A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__AND(isNat(z0), isNatList(x1)), A__ISNAT(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), cons(z0, z1)), A__AND(a__isNatList(cons(z0, z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) by A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(and(a__and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(and(a__and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(and(a__and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2 ---------------------------------------- (109) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c18_3, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__LENGTH(cons(x0, z0)) -> c18(A__U11(a__and(isNatList(z0), isNat(x0)), z0), A__AND(a__isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) by A__LENGTH(cons(x0, x1)) -> c18(A__U11(and(isNatList(x1), isNat(x0)), x1), A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__U11(and(isNatList(x1), isNat(x0)), x1), A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__U11(and(isNatList(x1), isNat(x0)), x1), A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3 ---------------------------------------- (113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) by A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0)), A__ISNATLIST(nil)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__AND(isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0)), A__ISNATLIST(nil)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__AND(isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0)), A__ISNATLIST(nil)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, z0)) -> c18(A__AND(isNatList(z0), isNat(x0)), A__ISNATLIST(z0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c18_2, c18_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__LENGTH(cons(x0, nil)) -> c18(A__AND(a__isNatList(nil), isNat(x0))) by A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(isNatList(nil), isNat(x0))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(isNatList(nil), isNat(x0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) A__LENGTH(cons(x0, nil)) -> c18(A__AND(isNatList(nil), isNat(x0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c18_2, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__LENGTH(cons(x0, nil)) -> c18(A__AND(isNatList(nil), isNat(x0))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(U11(z0, z1), x1)) -> c21(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) by MARK(U11(U11(x0, z1), x2)) -> c21(A__U11(U11(mark(x0), z1), x2), MARK(U11(x0, z1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2), MARK(U11(0, x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2), MARK(U11(nil, x1))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(x0, z1), x2)) -> c21(A__U11(U11(mark(x0), z1), x2), MARK(U11(x0, z1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2), MARK(U11(0, x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2), MARK(U11(nil, x1))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(x0, z1), x2)) -> c21(A__U11(U11(mark(x0), z1), x2), MARK(U11(x0, z1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2), MARK(U11(0, x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2), MARK(U11(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2 ---------------------------------------- (125) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(length(z0), x1)) -> c21(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) by MARK(U11(length(x0), x1)) -> c21(A__U11(length(mark(x0)), x1), MARK(length(x0))) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1), MARK(length(0))) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1), MARK(length(tt))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1), MARK(length(nil))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(x0), x1)) -> c21(A__U11(length(mark(x0)), x1), MARK(length(x0))) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1), MARK(length(0))) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1), MARK(length(tt))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1), MARK(length(nil))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(x0), x1)) -> c21(A__U11(length(mark(x0)), x1), MARK(length(x0))) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1), MARK(length(0))) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1), MARK(length(tt))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1), MARK(length(nil))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2 ---------------------------------------- (129) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(and(z0, z1), x1)) -> c21(A__U11(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) by MARK(U11(and(x0, z1), x2)) -> c21(A__U11(and(mark(x0), z1), x2), MARK(and(x0, z1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2), MARK(and(0, x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2), MARK(and(nil, x1))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(x0, z1), x2)) -> c21(A__U11(and(mark(x0), z1), x2), MARK(and(x0, z1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2), MARK(and(0, x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2), MARK(and(nil, x1))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(x0, z1), x2)) -> c21(A__U11(and(mark(x0), z1), x2), MARK(and(x0, z1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2), MARK(and(0, x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2), MARK(and(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2 ---------------------------------------- (133) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(isNat(z0), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(z0))) by MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(isNat(z0), x1), MARK(isNat(z0))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(isNat(z0), x1), MARK(isNat(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(A__U11(isNat(z0), x1), MARK(isNat(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2 ---------------------------------------- (137) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(isNatList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatList(z0))) by MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(isNatList(z0), x1), MARK(isNatList(z0))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(isNatList(z0), x1), MARK(isNatList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(A__U11(isNatList(z0), x1), MARK(isNatList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2 ---------------------------------------- (141) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatIList(z0), x1), MARK(isNatIList(z0))) by MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(isNatIList(z0), x1), MARK(isNatIList(z0))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(isNatIList(z0), x1), MARK(isNatIList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(isNatIList(z0), x1), MARK(isNatIList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (145) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (147) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(zeros, x1)) -> c21(A__U11(a__zeros, x1)) by MARK(U11(zeros, x0)) -> c21(A__U11(cons(0, zeros), x0)) MARK(U11(zeros, x0)) -> c21(A__U11(zeros, x0)) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) MARK(U11(zeros, x0)) -> c21(A__U11(cons(0, zeros), x0)) MARK(U11(zeros, x0)) -> c21(A__U11(zeros, x0)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) MARK(U11(zeros, x0)) -> c21(A__U11(cons(0, zeros), x0)) MARK(U11(zeros, x0)) -> c21(A__U11(zeros, x0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (149) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(U11(zeros, x0)) -> c21(A__U11(cons(0, zeros), x0)) MARK(U11(zeros, x0)) -> c21(A__U11(zeros, x0)) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (151) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (153) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(U11(z0, z1))) -> c22(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) by MARK(length(U11(x0, z1))) -> c22(A__LENGTH(U11(mark(x0), z1)), MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1)), MARK(U11(0, x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1)), MARK(U11(nil, x1))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(x0, z1))) -> c22(A__LENGTH(U11(mark(x0), z1)), MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1)), MARK(U11(0, x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1)), MARK(U11(nil, x1))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(x0, z1))) -> c22(A__LENGTH(U11(mark(x0), z1)), MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1)), MARK(U11(0, x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1)), MARK(U11(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (155) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (157) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(length(z0))) -> c22(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) by MARK(length(length(x0))) -> c22(A__LENGTH(length(mark(x0))), MARK(length(x0))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0)), MARK(length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt)), MARK(length(tt))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil)), MARK(length(nil))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(x0))) -> c22(A__LENGTH(length(mark(x0))), MARK(length(x0))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0)), MARK(length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt)), MARK(length(tt))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil)), MARK(length(nil))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(x0))) -> c22(A__LENGTH(length(mark(x0))), MARK(length(x0))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0)), MARK(length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt)), MARK(length(tt))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil)), MARK(length(nil))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (159) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (161) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(and(z0, z1))) -> c22(A__LENGTH(a__and(mark(z0), z1)), MARK(and(z0, z1))) by MARK(length(and(x0, z1))) -> c22(A__LENGTH(and(mark(x0), z1)), MARK(and(x0, z1))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1)), MARK(and(0, x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1)), MARK(and(nil, x1))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(x0, z1))) -> c22(A__LENGTH(and(mark(x0), z1)), MARK(and(x0, z1))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1)), MARK(and(0, x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1)), MARK(and(nil, x1))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(x0, z1))) -> c22(A__LENGTH(and(mark(x0), z1)), MARK(and(x0, z1))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1)), MARK(and(0, x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1)), MARK(and(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (163) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (165) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(isNat(z0))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(z0))) by MARK(length(isNat(0))) -> c22(A__LENGTH(tt), MARK(isNat(0))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(z0))) -> c22(A__LENGTH(isNat(z0)), MARK(isNat(z0))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(0))) -> c22(A__LENGTH(tt), MARK(isNat(0))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(z0))) -> c22(A__LENGTH(isNat(z0)), MARK(isNat(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(0))) -> c22(A__LENGTH(tt), MARK(isNat(0))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(z0))) -> c22(A__LENGTH(isNat(z0)), MARK(isNat(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (167) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (169) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(isNatList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatList(z0))) by MARK(length(isNatList(nil))) -> c22(A__LENGTH(tt), MARK(isNatList(nil))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(isNatList(z0)), MARK(isNatList(z0))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(nil))) -> c22(A__LENGTH(tt), MARK(isNatList(nil))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(isNatList(z0)), MARK(isNatList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(nil))) -> c22(A__LENGTH(tt), MARK(isNatList(nil))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(z0))) -> c22(A__LENGTH(isNatList(z0)), MARK(isNatList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (171) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (173) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatIList(z0)), MARK(isNatIList(z0))) by MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(zeros))) -> c22(A__LENGTH(tt), MARK(isNatIList(zeros))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(isNatIList(z0)), MARK(isNatIList(z0))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(zeros))) -> c22(A__LENGTH(tt), MARK(isNatIList(zeros))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(isNatIList(z0)), MARK(isNatIList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(zeros))) -> c22(A__LENGTH(tt), MARK(isNatIList(zeros))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(isNatIList(z0)), MARK(isNatIList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (175) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (177) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(zeros)) -> c22(A__LENGTH(a__zeros)) by MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(length(zeros)) -> c22(A__LENGTH(zeros)) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(length(zeros)) -> c22(A__LENGTH(zeros)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(length(zeros)) -> c22(A__LENGTH(zeros)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (179) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(length(zeros)) -> c22(A__LENGTH(zeros)) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (181) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(and(U11(z0, z1), x1)) -> c23(A__AND(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) by MARK(and(U11(x0, z1), x2)) -> c23(A__AND(U11(mark(x0), z1), x2), MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2), MARK(U11(0, x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2), MARK(U11(nil, x1))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(x0, z1), x2)) -> c23(A__AND(U11(mark(x0), z1), x2), MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2), MARK(U11(0, x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2), MARK(U11(nil, x1))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), 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c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(x0, z1), x2)) -> c23(A__AND(U11(mark(x0), z1), x2), MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2), MARK(U11(0, x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2), MARK(U11(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (183) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, 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MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> 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MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (185) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(and(length(z0), x1)) -> c23(A__AND(a__length(mark(z0)), x1), MARK(length(z0))) by MARK(and(length(x0), x1)) -> c23(A__AND(length(mark(x0)), x1), MARK(length(x0))) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1), MARK(length(0))) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1), MARK(length(tt))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1), MARK(length(nil))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), 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MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, 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c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(x0), x1)) -> c23(A__AND(length(mark(x0)), x1), MARK(length(x0))) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1), MARK(length(0))) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1), MARK(length(tt))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1), MARK(length(nil))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (187) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), 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x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (189) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(and(and(z0, z1), x1)) -> c23(A__AND(a__and(mark(z0), z1), x1), MARK(and(z0, z1))) by MARK(and(and(x0, z1), x2)) -> c23(A__AND(and(mark(x0), z1), x2), MARK(and(x0, z1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2), MARK(and(0, x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2), MARK(and(nil, x1))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(x0, z1), x2)) -> c23(A__AND(and(mark(x0), z1), x2), MARK(and(x0, z1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2), MARK(and(0, x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2), MARK(and(nil, x1))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(x0, z1), x2)) -> c23(A__AND(and(mark(x0), z1), x2), MARK(and(x0, z1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2), MARK(and(0, x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2), MARK(and(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (191) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (193) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(and(isNat(z0), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(z0))) by MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(A__AND(isNat(z0), x1), MARK(isNat(z0))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(A__AND(isNat(z0), x1), MARK(isNat(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> 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c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(A__AND(isNat(z0), x1), MARK(isNat(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (195) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (197) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(and(isNatList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatList(z0))) by MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(isNatList(z0), x1), MARK(isNatList(z0))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, 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MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(isNatList(z0), x1), MARK(isNatList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) 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MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, 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MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(A__AND(isNatList(z0), x1), MARK(isNatList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (199) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_2, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2 ---------------------------------------- (201) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatIList(z0), x1), MARK(isNatIList(z0))) by MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(isNatIList(z0), x1), MARK(isNatIList(z0))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(isNatIList(z0), x1), MARK(isNatIList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(isNatIList(z0), x1), MARK(isNatIList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2, c23_2 ---------------------------------------- (203) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> 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A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) 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MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2, c23_2 ---------------------------------------- (205) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(and(zeros, x1)) -> c23(A__AND(a__zeros, x1)) by MARK(and(zeros, x0)) -> c23(A__AND(cons(0, zeros), x0)) MARK(and(zeros, x0)) -> c23(A__AND(zeros, x0)) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, 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MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, 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z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) MARK(and(zeros, x0)) -> c23(A__AND(cons(0, zeros), x0)) MARK(and(zeros, x0)) -> c23(A__AND(zeros, x0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2, c23_2 ---------------------------------------- (207) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(and(zeros, x0)) -> c23(A__AND(zeros, x0)) MARK(and(zeros, x0)) -> c23(A__AND(cons(0, zeros), x0)) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, 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x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1)), MARK(and(zeros, x1))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2), MARK(and(zeros, x1))) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2, c23_2 ---------------------------------------- (209) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), 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MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), 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x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2)) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2)) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2, c23_2 ---------------------------------------- (211) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U11(tt, U11(0, x1)) -> c2(A__LENGTH(a__U11(0, x1))) by A__U11(tt, U11(0, z1)) -> c2(A__LENGTH(U11(0, z1))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> 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MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2)) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2)) A__U11(tt, U11(0, z1)) -> c2(A__LENGTH(U11(0, z1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2, c23_2 ---------------------------------------- (213) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U11(tt, U11(0, z1)) -> c2(A__LENGTH(U11(0, z1))) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> 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x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2)) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2)) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2, c23_2 ---------------------------------------- (215) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U11(tt, U11(nil, x1)) -> c2(A__LENGTH(a__U11(nil, x1))) by A__U11(tt, U11(nil, z1)) -> c2(A__LENGTH(U11(nil, z1))) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), 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c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2)) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2)) A__U11(tt, U11(nil, z1)) -> c2(A__LENGTH(U11(nil, z1))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2)) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2)) A__U11(tt, U11(nil, z1)) -> c2(A__LENGTH(U11(nil, z1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2, c23_2 ---------------------------------------- (217) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U11(tt, U11(nil, z1)) -> c2(A__LENGTH(U11(nil, z1))) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, 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MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> 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c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> 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x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2)) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), 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MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2)) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2, c23_2 ---------------------------------------- (219) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U11(tt, length(0)) -> c2(A__LENGTH(a__length(0))) by A__U11(tt, length(0)) -> c2(A__LENGTH(length(0))) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2)) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2)) A__U11(tt, length(0)) -> c2(A__LENGTH(length(0))) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2)) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2)) A__U11(tt, length(0)) -> c2(A__LENGTH(length(0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2, c23_2 ---------------------------------------- (221) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U11(tt, length(0)) -> c2(A__LENGTH(length(0))) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> s(a__length(mark(z0))) a__U11(z0, z1) -> U11(z0, z1) a__and(tt, z0) -> mark(z0) a__and(z0, z1) -> and(z0, z1) a__isNat(0) -> tt a__isNat(length(z0)) -> a__isNatList(z0) a__isNat(s(z0)) -> a__isNat(z0) a__isNat(z0) -> isNat(z0) a__isNatIList(z0) -> a__isNatList(z0) a__isNatIList(zeros) -> tt a__isNatIList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatIList(z1)) a__isNatIList(z0) -> isNatIList(z0) a__isNatList(nil) -> tt a__isNatList(cons(z0, z1)) -> a__and(a__isNat(z0), isNatList(z1)) a__isNatList(z0) -> isNatList(z0) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(a__and(a__isNatList(z1), isNat(z0)), z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(and(z0, z1)) -> a__and(mark(z0), z1) mark(isNat(z0)) -> a__isNat(z0) mark(isNatList(z0)) -> a__isNatList(z0) mark(isNatIList(z0)) -> a__isNatIList(z0) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, x1)) -> c18(A__AND(a__isNatList(x1), isNat(x0)), A__ISNATLIST(x1)) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2)) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2)) S tuples: A__AND(tt, z0) -> c4(MARK(z0)) A__ISNAT(length(z0)) -> c7(A__ISNATLIST(z0)) A__ISNAT(s(z0)) -> c8(A__ISNAT(z0)) A__ISNATILIST(z0) -> c10(A__ISNATLIST(z0)) MARK(isNat(z0)) -> c24(A__ISNAT(z0)) MARK(isNatList(z0)) -> c25(A__ISNATLIST(z0)) MARK(isNatIList(z0)) -> c26(A__ISNATILIST(z0)) MARK(cons(z0, z1)) -> c27(MARK(z0)) MARK(s(z0)) -> c30(MARK(z0)) A__U11(tt, cons(z0, z1)) -> c2(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U11(tt, s(z0)) -> c2(MARK(s(z0))) A__ISNATILIST(cons(0, x1)) -> c12(A__AND(tt, isNatIList(x1))) A__ISNATILIST(cons(z0, x1)) -> c12(A__ISNAT(z0)) A__ISNATLIST(cons(0, x1)) -> c15(A__AND(tt, isNatList(x1))) A__ISNATLIST(cons(z0, x1)) -> c15(A__ISNAT(z0)) MARK(U11(cons(z0, z1), x1)) -> c21(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c21(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c21(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c22(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c22(MARK(s(z0))) MARK(and(cons(z0, z1), x1)) -> c23(MARK(cons(z0, z1))) MARK(and(tt, x1)) -> c23(A__AND(tt, x1)) MARK(and(s(z0), x1)) -> c23(MARK(s(z0))) A__U11(tt, U11(U11(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U11(tt, U11(length(z0), x1)) -> c2(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U11(tt, U11(and(z0, z1), x1)) -> c2(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) A__U11(tt, U11(isNat(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) A__U11(tt, U11(isNatList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) A__U11(tt, U11(isNatIList(z0), x1)) -> c2(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) A__U11(tt, U11(cons(z0, z1), x1)) -> c2(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U11(tt, U11(tt, x1)) -> c2(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U11(tt, U11(s(z0), x1)) -> c2(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U11(tt, U11(x0, z1)) -> c2(MARK(U11(x0, z1))) A__U11(tt, length(zeros)) -> c2(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U11(tt, length(U11(z0, z1))) -> c2(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U11(tt, length(length(z0))) -> c2(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U11(tt, length(and(z0, z1))) -> c2(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) A__U11(tt, length(isNat(z0))) -> c2(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) A__U11(tt, length(isNatList(z0))) -> c2(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) A__U11(tt, length(isNatIList(z0))) -> c2(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) A__U11(tt, length(cons(z0, z1))) -> c2(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U11(tt, length(s(z0))) -> c2(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U11(tt, length(x0)) -> c2(MARK(length(x0))) A__U11(tt, length(tt)) -> c2(A__LENGTH(a__length(tt))) A__U11(tt, length(nil)) -> c2(A__LENGTH(a__length(nil))) A__U11(tt, and(U11(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) A__U11(tt, and(length(z0), x1)) -> c2(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) A__U11(tt, and(and(z0, z1), x1)) -> c2(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) A__U11(tt, and(isNat(z0), x1)) -> c2(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) A__U11(tt, and(isNatList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) A__U11(tt, and(isNatIList(z0), x1)) -> c2(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) A__U11(tt, and(cons(z0, z1), x1)) -> c2(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) A__U11(tt, and(tt, x1)) -> c2(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) A__U11(tt, and(s(z0), x1)) -> c2(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) A__U11(tt, and(x0, z1)) -> c2(MARK(and(x0, z1))) A__U11(tt, and(0, x1)) -> c2(A__LENGTH(a__and(0, x1))) A__U11(tt, and(nil, x1)) -> c2(A__LENGTH(a__and(nil, x1))) A__U11(tt, isNat(length(z0))) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) A__U11(tt, isNat(s(z0))) -> c2(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) A__U11(tt, isNat(0)) -> c2(MARK(isNat(0))) A__U11(tt, isNat(z0)) -> c2(MARK(isNat(z0))) A__U11(tt, isNatList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) A__U11(tt, isNatList(nil)) -> c2(MARK(isNatList(nil))) A__U11(tt, isNatList(z0)) -> c2(MARK(isNatList(z0))) A__U11(tt, isNatIList(z0)) -> c2(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) A__U11(tt, isNatIList(cons(z0, z1))) -> c2(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) A__U11(tt, isNatIList(zeros)) -> c2(MARK(isNatIList(zeros))) A__U11(tt, isNatIList(z0)) -> c2(MARK(isNatIList(z0))) A__U11(tt, zeros) -> c2(A__LENGTH(cons(0, zeros))) A__ISNATILIST(cons(length(nil), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(length(nil))) A__ISNATILIST(cons(length(cons(z0, z1)), x1)) -> c12(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatIList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATILIST(cons(length(z0), x1)) -> c12(A__ISNAT(length(z0))) A__ISNATILIST(cons(s(0), x1)) -> c12(A__AND(tt, isNatIList(x1)), A__ISNAT(s(0))) A__ISNATILIST(cons(s(length(z0)), x1)) -> c12(A__AND(a__isNatList(z0), isNatIList(x1)), A__ISNAT(s(length(z0)))) A__ISNATILIST(cons(s(s(z0)), x1)) -> c12(A__AND(a__isNat(z0), isNatIList(x1)), A__ISNAT(s(s(z0)))) A__ISNATILIST(cons(s(z0), x1)) -> c12(A__ISNAT(s(z0))) A__ISNATLIST(cons(length(nil), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(length(nil))) A__ISNATLIST(cons(length(cons(z0, z1)), x1)) -> c15(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNatList(x1)), A__ISNAT(length(cons(z0, z1)))) A__ISNATLIST(cons(length(z0), x1)) -> c15(A__ISNAT(length(z0))) A__ISNATLIST(cons(s(0), x1)) -> c15(A__AND(tt, isNatList(x1)), A__ISNAT(s(0))) A__ISNATLIST(cons(s(length(z0)), x1)) -> c15(A__AND(a__isNatList(z0), isNatList(x1)), A__ISNAT(s(length(z0)))) A__ISNATLIST(cons(s(s(z0)), x1)) -> c15(A__AND(a__isNat(z0), isNatList(x1)), A__ISNAT(s(s(z0)))) A__ISNATLIST(cons(s(z0), x1)) -> c15(A__ISNAT(s(z0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__U11(a__and(and(a__isNat(x1), isNatList(x2)), isNat(x0)), cons(x1, x2)), A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(0, x2))) -> c18(A__U11(a__and(a__and(tt, isNatList(x2)), isNat(x0)), cons(0, x2)), A__AND(a__isNatList(cons(0, x2)), isNat(x0)), A__ISNATLIST(cons(0, x2))) A__LENGTH(cons(x0, cons(length(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNatList(z0), isNatList(x2)), isNat(x0)), cons(length(z0), x2)), A__AND(a__isNatList(cons(length(z0), x2)), isNat(x0)), A__ISNATLIST(cons(length(z0), x2))) A__LENGTH(cons(x0, cons(s(z0), x2))) -> c18(A__U11(a__and(a__and(a__isNat(z0), isNatList(x2)), isNat(x0)), cons(s(z0), x2)), A__AND(a__isNatList(cons(s(z0), x2)), isNat(x0)), A__ISNATLIST(cons(s(z0), x2))) A__LENGTH(cons(x0, cons(z0, x2))) -> c18(A__U11(a__and(a__and(isNat(z0), isNatList(x2)), isNat(x0)), cons(z0, x2)), A__AND(a__isNatList(cons(z0, x2)), isNat(x0)), A__ISNATLIST(cons(z0, x2))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0))) A__LENGTH(cons(x0, cons(x1, x2))) -> c18(A__AND(a__isNatList(cons(x1, x2)), isNat(x0)), A__ISNATLIST(cons(x1, x2))) A__LENGTH(cons(x0, cons(z0, z1))) -> c18(A__AND(a__and(a__isNat(z0), isNatList(z1)), isNat(x0)), A__ISNATLIST(cons(z0, z1))) A__LENGTH(cons(x0, nil)) -> c18(A__AND(tt, isNat(x0))) A__LENGTH(cons(x0, z0)) -> c18(A__ISNATLIST(z0)) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c21(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(and(z0, z1), x1), x2)) -> c21(A__U11(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(U11(U11(isNat(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(U11(U11(isNatList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(U11(U11(isNatIList(z0), x1), x2)) -> c21(A__U11(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c21(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c21(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c21(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c21(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c21(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c21(A__U11(a__U11(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c21(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c21(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c21(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(and(z0, z1)), x1)) -> c21(A__U11(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(U11(length(isNat(z0)), x1)) -> c21(A__U11(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(U11(length(isNatList(z0)), x1)) -> c21(A__U11(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(U11(length(isNatIList(z0)), x1)) -> c21(A__U11(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c21(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c21(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c21(MARK(length(x0))) MARK(U11(length(0), x1)) -> c21(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c21(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c21(A__U11(a__length(nil), x1)) MARK(U11(and(U11(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(U11(and(length(z0), x1), x2)) -> c21(A__U11(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(U11(and(and(z0, z1), x1), x2)) -> c21(A__U11(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(U11(and(isNat(z0), x1), x2)) -> c21(A__U11(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(U11(and(isNatList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(U11(and(isNatIList(z0), x1), x2)) -> c21(A__U11(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(U11(and(cons(z0, z1), x1), x2)) -> c21(A__U11(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(U11(and(tt, x1), x2)) -> c21(A__U11(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(U11(and(s(z0), x1), x2)) -> c21(A__U11(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(U11(and(x0, z1), x2)) -> c21(MARK(and(x0, z1))) MARK(U11(and(0, x1), x2)) -> c21(A__U11(a__and(0, x1), x2)) MARK(U11(and(nil, x1), x2)) -> c21(A__U11(a__and(nil, x1), x2)) MARK(U11(isNat(0), x1)) -> c21(A__U11(tt, x1), MARK(isNat(0))) MARK(U11(isNat(length(z0)), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(U11(isNat(s(z0)), x1)) -> c21(A__U11(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(U11(isNat(z0), x1)) -> c21(MARK(isNat(z0))) MARK(U11(isNatList(nil), x1)) -> c21(A__U11(tt, x1), MARK(isNatList(nil))) MARK(U11(isNatList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(U11(isNatList(z0), x1)) -> c21(MARK(isNatList(z0))) MARK(U11(isNatIList(z0), x1)) -> c21(A__U11(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(U11(isNatIList(zeros), x1)) -> c21(A__U11(tt, x1), MARK(isNatIList(zeros))) MARK(U11(isNatIList(cons(z0, z1)), x1)) -> c21(A__U11(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(U11(isNatIList(z0), x1)) -> c21(MARK(isNatIList(z0))) A__U11(tt, U11(zeros, x1)) -> c2(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c21(A__U11(a__U11(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c22(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(and(z0, z1), x1))) -> c22(A__LENGTH(a__U11(a__and(mark(z0), z1), x1)), MARK(U11(and(z0, z1), x1))) MARK(length(U11(isNat(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNat(z0), x1)), MARK(U11(isNat(z0), x1))) MARK(length(U11(isNatList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatList(z0), x1)), MARK(U11(isNatList(z0), x1))) MARK(length(U11(isNatIList(z0), x1))) -> c22(A__LENGTH(a__U11(a__isNatIList(z0), x1)), MARK(U11(isNatIList(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c22(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c22(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c22(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c22(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c22(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c22(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c22(A__LENGTH(a__U11(nil, x1))) MARK(length(length(zeros))) -> c22(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c22(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(length(z0)))) -> c22(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(and(z0, z1)))) -> c22(A__LENGTH(a__length(a__and(mark(z0), z1))), MARK(length(and(z0, z1)))) MARK(length(length(isNat(z0)))) -> c22(A__LENGTH(a__length(a__isNat(z0))), MARK(length(isNat(z0)))) MARK(length(length(isNatList(z0)))) -> c22(A__LENGTH(a__length(a__isNatList(z0))), MARK(length(isNatList(z0)))) MARK(length(length(isNatIList(z0)))) -> c22(A__LENGTH(a__length(a__isNatIList(z0))), MARK(length(isNatIList(z0)))) MARK(length(length(cons(z0, z1)))) -> c22(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c22(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c22(MARK(length(x0))) MARK(length(length(0))) -> c22(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c22(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c22(A__LENGTH(a__length(nil))) MARK(length(and(U11(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__U11(mark(z0), z1), x1)), MARK(and(U11(z0, z1), x1))) MARK(length(and(length(z0), x1))) -> c22(A__LENGTH(a__and(a__length(mark(z0)), x1)), MARK(and(length(z0), x1))) MARK(length(and(and(z0, z1), x1))) -> c22(A__LENGTH(a__and(a__and(mark(z0), z1), x1)), MARK(and(and(z0, z1), x1))) MARK(length(and(isNat(z0), x1))) -> c22(A__LENGTH(a__and(a__isNat(z0), x1)), MARK(and(isNat(z0), x1))) MARK(length(and(isNatList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatList(z0), x1)), MARK(and(isNatList(z0), x1))) MARK(length(and(isNatIList(z0), x1))) -> c22(A__LENGTH(a__and(a__isNatIList(z0), x1)), MARK(and(isNatIList(z0), x1))) MARK(length(and(cons(z0, z1), x1))) -> c22(A__LENGTH(a__and(cons(mark(z0), z1), x1)), MARK(and(cons(z0, z1), x1))) MARK(length(and(tt, x1))) -> c22(A__LENGTH(a__and(tt, x1)), MARK(and(tt, x1))) MARK(length(and(s(z0), x1))) -> c22(A__LENGTH(a__and(s(mark(z0)), x1)), MARK(and(s(z0), x1))) MARK(length(and(x0, z1))) -> c22(MARK(and(x0, z1))) MARK(length(and(0, x1))) -> c22(A__LENGTH(a__and(0, x1))) MARK(length(and(nil, x1))) -> c22(A__LENGTH(a__and(nil, x1))) MARK(length(isNat(length(z0)))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNat(length(z0)))) MARK(length(isNat(s(z0)))) -> c22(A__LENGTH(a__isNat(z0)), MARK(isNat(s(z0)))) MARK(length(isNat(0))) -> c22(MARK(isNat(0))) MARK(length(isNat(z0))) -> c22(MARK(isNat(z0))) MARK(length(isNatList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatList(z1))), MARK(isNatList(cons(z0, z1)))) MARK(length(isNatList(nil))) -> c22(MARK(isNatList(nil))) MARK(length(isNatList(z0))) -> c22(MARK(isNatList(z0))) MARK(length(isNatIList(z0))) -> c22(A__LENGTH(a__isNatList(z0)), MARK(isNatIList(z0))) MARK(length(isNatIList(cons(z0, z1)))) -> c22(A__LENGTH(a__and(a__isNat(z0), isNatIList(z1))), MARK(isNatIList(cons(z0, z1)))) MARK(length(isNatIList(zeros))) -> c22(MARK(isNatIList(zeros))) MARK(length(isNatIList(z0))) -> c22(MARK(isNatIList(z0))) MARK(length(zeros)) -> c22(A__LENGTH(cons(0, zeros))) MARK(and(U11(U11(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(and(U11(length(z0), x1), x2)) -> c23(A__AND(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(and(U11(and(z0, z1), x1), x2)) -> c23(A__AND(a__U11(a__and(mark(z0), z1), x1), x2), MARK(U11(and(z0, z1), x1))) MARK(and(U11(isNat(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNat(z0), x1), x2), MARK(U11(isNat(z0), x1))) MARK(and(U11(isNatList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatList(z0), x1), x2), MARK(U11(isNatList(z0), x1))) MARK(and(U11(isNatIList(z0), x1), x2)) -> c23(A__AND(a__U11(a__isNatIList(z0), x1), x2), MARK(U11(isNatIList(z0), x1))) MARK(and(U11(cons(z0, z1), x1), x2)) -> c23(A__AND(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(and(U11(tt, x1), x2)) -> c23(A__AND(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(and(U11(s(z0), x1), x2)) -> c23(A__AND(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(and(U11(x0, z1), x2)) -> c23(MARK(U11(x0, z1))) MARK(and(U11(zeros, x1), x2)) -> c23(A__AND(a__U11(a__zeros, x1), x2)) MARK(and(U11(0, x1), x2)) -> c23(A__AND(a__U11(0, x1), x2)) MARK(and(U11(nil, x1), x2)) -> c23(A__AND(a__U11(nil, x1), x2)) MARK(and(length(zeros), x1)) -> c23(A__AND(a__length(a__zeros), x1), MARK(length(zeros))) MARK(and(length(U11(z0, z1)), x1)) -> c23(A__AND(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(and(length(length(z0)), x1)) -> c23(A__AND(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(and(length(and(z0, z1)), x1)) -> c23(A__AND(a__length(a__and(mark(z0), z1)), x1), MARK(length(and(z0, z1)))) MARK(and(length(isNat(z0)), x1)) -> c23(A__AND(a__length(a__isNat(z0)), x1), MARK(length(isNat(z0)))) MARK(and(length(isNatList(z0)), x1)) -> c23(A__AND(a__length(a__isNatList(z0)), x1), MARK(length(isNatList(z0)))) MARK(and(length(isNatIList(z0)), x1)) -> c23(A__AND(a__length(a__isNatIList(z0)), x1), MARK(length(isNatIList(z0)))) MARK(and(length(cons(z0, z1)), x1)) -> c23(A__AND(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(and(length(s(z0)), x1)) -> c23(A__AND(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(and(length(x0), x1)) -> c23(MARK(length(x0))) MARK(and(length(0), x1)) -> c23(A__AND(a__length(0), x1)) MARK(and(length(tt), x1)) -> c23(A__AND(a__length(tt), x1)) MARK(and(length(nil), x1)) -> c23(A__AND(a__length(nil), x1)) MARK(and(and(U11(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__U11(mark(z0), z1), x1), x2), MARK(and(U11(z0, z1), x1))) MARK(and(and(length(z0), x1), x2)) -> c23(A__AND(a__and(a__length(mark(z0)), x1), x2), MARK(and(length(z0), x1))) MARK(and(and(and(z0, z1), x1), x2)) -> c23(A__AND(a__and(a__and(mark(z0), z1), x1), x2), MARK(and(and(z0, z1), x1))) MARK(and(and(isNat(z0), x1), x2)) -> c23(A__AND(a__and(a__isNat(z0), x1), x2), MARK(and(isNat(z0), x1))) MARK(and(and(isNatList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatList(z0), x1), x2), MARK(and(isNatList(z0), x1))) MARK(and(and(isNatIList(z0), x1), x2)) -> c23(A__AND(a__and(a__isNatIList(z0), x1), x2), MARK(and(isNatIList(z0), x1))) MARK(and(and(cons(z0, z1), x1), x2)) -> c23(A__AND(a__and(cons(mark(z0), z1), x1), x2), MARK(and(cons(z0, z1), x1))) MARK(and(and(tt, x1), x2)) -> c23(A__AND(a__and(tt, x1), x2), MARK(and(tt, x1))) MARK(and(and(s(z0), x1), x2)) -> c23(A__AND(a__and(s(mark(z0)), x1), x2), MARK(and(s(z0), x1))) MARK(and(and(x0, z1), x2)) -> c23(MARK(and(x0, z1))) MARK(and(and(0, x1), x2)) -> c23(A__AND(a__and(0, x1), x2)) MARK(and(and(nil, x1), x2)) -> c23(A__AND(a__and(nil, x1), x2)) MARK(and(isNat(0), x1)) -> c23(A__AND(tt, x1), MARK(isNat(0))) MARK(and(isNat(length(z0)), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNat(length(z0)))) MARK(and(isNat(s(z0)), x1)) -> c23(A__AND(a__isNat(z0), x1), MARK(isNat(s(z0)))) MARK(and(isNat(z0), x1)) -> c23(MARK(isNat(z0))) MARK(and(isNatList(nil), x1)) -> c23(A__AND(tt, x1), MARK(isNatList(nil))) MARK(and(isNatList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatList(z1)), x1), MARK(isNatList(cons(z0, z1)))) MARK(and(isNatList(z0), x1)) -> c23(MARK(isNatList(z0))) MARK(and(isNatIList(z0), x1)) -> c23(A__AND(a__isNatList(z0), x1), MARK(isNatIList(z0))) MARK(and(isNatIList(zeros), x1)) -> c23(A__AND(tt, x1), MARK(isNatIList(zeros))) MARK(and(isNatIList(cons(z0, z1)), x1)) -> c23(A__AND(a__and(a__isNat(z0), isNatIList(z1)), x1), MARK(isNatIList(cons(z0, z1)))) MARK(and(isNatIList(z0), x1)) -> c23(MARK(isNatIList(z0))) A__U11(tt, and(zeros, x1)) -> c2(A__LENGTH(a__and(a__zeros, x1))) MARK(U11(and(zeros, x1), x2)) -> c21(A__U11(a__and(a__zeros, x1), x2)) MARK(length(and(zeros, x1))) -> c22(A__LENGTH(a__and(a__zeros, x1))) MARK(and(and(zeros, x1), x2)) -> c23(A__AND(a__and(a__zeros, x1), x2)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__and_2, a__isNat_1, a__isNatIList_1, a__isNatList_1, a__length_1, mark_1 Defined Pair Symbols: A__AND_2, A__ISNAT_1, A__ISNATILIST_1, MARK_1, A__U11_2, A__ISNATLIST_1, A__LENGTH_1 Compound Symbols: c4_1, c7_1, c8_1, c10_1, c24_1, c25_1, c26_1, c27_1, c30_1, c2_2, c2_1, c12_1, c15_1, c21_1, c22_2, c22_1, c23_1, c12_2, c15_2, c18_3, c18_1, c18_2, c21_2, c23_2