KILLED proof of input_WAjgIgtkM9.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (6) CdtProblem (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxRelTRS (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxRelTRS (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (12) typed CpxTrs (13) OrderProof [LOWER BOUND(ID), 0 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 2302 ms] (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 3901 ms] (18) BEST (19) proven lower bound (20) LowerBoundPropagationProof [FINISHED, 0 ms] (21) BOUNDS(n^1, INF) (22) typed CpxTrs (23) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (24) CdtProblem (25) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CpxRelTRS (29) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (30) CpxTRS (31) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CpxWeightedTrs (33) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CpxTypedWeightedTrs (35) CompletionProof [UPPER BOUND(ID), 0 ms] (36) CpxTypedWeightedCompleteTrs (37) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CpxTypedWeightedCompleteTrs (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) InliningProof [UPPER BOUND(ID), 632 ms] (42) CpxRNTS (43) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CpxRNTS (45) CompletionProof [UPPER BOUND(ID), 0 ms] (46) CpxTypedWeightedCompleteTrs (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 5 ms] (90) CdtProblem (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 5 ms] (94) CdtProblem (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 9 ms] (108) CdtProblem (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms] (112) CdtProblem (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 49 ms] (142) CdtProblem (143) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 28 ms] (146) CdtProblem (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 105 ms] (150) CdtProblem (151) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 120 ms] (154) CdtProblem (155) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 199 ms] (158) CdtProblem (159) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 193 ms] (162) CdtProblem (163) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 187 ms] (166) CdtProblem (167) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 218 ms] (170) CdtProblem (171) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 189 ms] (174) CdtProblem (175) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 293 ms] (178) CdtProblem (179) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 288 ms] (182) CdtProblem (183) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 299 ms] (186) CdtProblem (187) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 250 ms] (190) CdtProblem (191) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 1 ms] (192) CdtProblem (193) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 283 ms] (194) CdtProblem (195) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 400 ms] (198) CdtProblem (199) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 385 ms] (202) CdtProblem (203) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 413 ms] (206) CdtProblem (207) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 390 ms] (210) CdtProblem (211) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 508 ms] (214) CdtProblem (215) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 484 ms] (218) CdtProblem (219) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (220) CdtProblem (221) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 513 ms] (222) CdtProblem (223) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CdtProblem (225) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 524 ms] (226) CdtProblem (227) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (228) CdtProblem (229) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 554 ms] (230) CdtProblem (231) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (232) CdtProblem (233) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 540 ms] (234) CdtProblem (235) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (236) CdtProblem (237) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 584 ms] (238) CdtProblem (239) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (240) CdtProblem (241) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 610 ms] (242) CdtProblem (243) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (244) CdtProblem (245) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (246) CpxWeightedTrs (247) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (248) CpxTypedWeightedTrs (249) CompletionProof [UPPER BOUND(ID), 0 ms] (250) CpxTypedWeightedCompleteTrs (251) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (252) CpxTypedWeightedCompleteTrs (253) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (254) CpxRNTS (255) InliningProof [UPPER BOUND(ID), 260 ms] (256) CpxRNTS (257) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (258) CpxRNTS (259) CompletionProof [UPPER BOUND(ID), 0 ms] (260) CpxTypedWeightedCompleteTrs (261) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (262) CpxRNTS ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(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__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(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) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(nil) -> 0' a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(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__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(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) -> a__U12(tt, L) a__U12(tt, L) -> s(a__length(mark(L))) a__length(nil) -> 0 a__length(cons(N, L)) -> a__U11(tt, L) mark(zeros) -> a__zeros mark(U11(X1, X2)) -> a__U11(mark(X1), X2) mark(U12(X1, X2)) -> a__U12(mark(X1), X2) mark(length(X)) -> a__length(mark(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__U12(X1, X2) -> U12(X1, X2) a__length(X) -> length(X) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__U11(z0, z1) -> c3 A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) A__U12(z0, z1) -> c5 A__LENGTH(nil) -> c6 A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) A__LENGTH(z0) -> c8 MARK(zeros) -> c9(A__ZEROS) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(0) -> c14 MARK(tt) -> c15 MARK(s(z0)) -> c16(MARK(z0)) MARK(nil) -> c17 S tuples: A__ZEROS -> c A__ZEROS -> c1 A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__U11(z0, z1) -> c3 A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) A__U12(z0, z1) -> c5 A__LENGTH(nil) -> c6 A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) A__LENGTH(z0) -> c8 MARK(zeros) -> c9(A__ZEROS) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(0) -> c14 MARK(tt) -> c15 MARK(s(z0)) -> c16(MARK(z0)) MARK(nil) -> c17 K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__ZEROS, A__U11_2, A__U12_2, A__LENGTH_1, MARK_1 Compound Symbols: c, c1, c2_1, c3, c4_2, c5, c6, c7_1, c8, c9_1, c10_2, c11_2, c12_2, c13_1, c14, c15, c16_1, c17 ---------------------------------------- (7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__ZEROS -> c A__ZEROS -> c1 A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__U11(z0, z1) -> c3 A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) A__U12(z0, z1) -> c5 A__LENGTH(nil) -> c6 A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) A__LENGTH(z0) -> c8 MARK(zeros) -> c9(A__ZEROS) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(0) -> c14 MARK(tt) -> c15 MARK(s(z0)) -> c16(MARK(z0)) MARK(nil) -> c17 The (relative) TRS S consists of the following rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Rewrite Strategy: INNERMOST ---------------------------------------- (9) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (10) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__ZEROS -> c A__ZEROS -> c1 A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__U11(z0, z1) -> c3 A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) A__U12(z0, z1) -> c5 A__LENGTH(nil) -> c6 A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) A__LENGTH(z0) -> c8 MARK(zeros) -> c9(A__ZEROS) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(0') -> c14 MARK(tt) -> c15 MARK(s(z0)) -> c16(MARK(z0)) MARK(nil) -> c17 The (relative) TRS S consists of the following rules: a__zeros -> cons(0', zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0' a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0') -> 0' mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (12) Obligation: Innermost TRS: Rules: A__ZEROS -> c A__ZEROS -> c1 A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__U11(z0, z1) -> c3 A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) A__U12(z0, z1) -> c5 A__LENGTH(nil) -> c6 A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) A__LENGTH(z0) -> c8 MARK(zeros) -> c9(A__ZEROS) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(0') -> c14 MARK(tt) -> c15 MARK(s(z0)) -> c16(MARK(z0)) MARK(nil) -> c17 a__zeros -> cons(0', zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0' a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0') -> 0' mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Types: A__ZEROS :: c:c1 c :: c:c1 c1 :: c:c1 A__U11 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> c2:c3 tt :: tt:nil:cons:zeros:U11:U12:length:0':s c2 :: c4:c5 -> c2:c3 A__U12 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> c4:c5 c3 :: c2:c3 c4 :: c6:c7:c8 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c4:c5 A__LENGTH :: tt:nil:cons:zeros:U11:U12:length:0':s -> c6:c7:c8 mark :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s MARK :: tt:nil:cons:zeros:U11:U12:length:0':s -> c9:c10:c11:c12:c13:c14:c15:c16:c17 c5 :: c4:c5 nil :: tt:nil:cons:zeros:U11:U12:length:0':s c6 :: c6:c7:c8 cons :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c7 :: c2:c3 -> c6:c7:c8 c8 :: c6:c7:c8 zeros :: tt:nil:cons:zeros:U11:U12:length:0':s c9 :: c:c1 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 U11 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c10 :: c2:c3 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 U12 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c11 :: c4:c5 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 length :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c12 :: c6:c7:c8 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 c13 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 0' :: tt:nil:cons:zeros:U11:U12:length:0':s c14 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 c15 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 s :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c16 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 c17 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 a__zeros :: tt:nil:cons:zeros:U11:U12:length:0':s a__U11 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s a__U12 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s a__length :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s hole_c:c11_18 :: c:c1 hole_c2:c32_18 :: c2:c3 hole_tt:nil:cons:zeros:U11:U12:length:0':s3_18 :: tt:nil:cons:zeros:U11:U12:length:0':s hole_c4:c54_18 :: c4:c5 hole_c6:c7:c85_18 :: c6:c7:c8 hole_c9:c10:c11:c12:c13:c14:c15:c16:c176_18 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18 :: Nat -> tt:nil:cons:zeros:U11:U12:length:0':s gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18 :: Nat -> c9:c10:c11:c12:c13:c14:c15:c16:c17 ---------------------------------------- (13) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: A__U11, A__U12, A__LENGTH, mark, MARK, a__U11, a__U12, a__length They will be analysed ascendingly in the following order: A__U11 = A__U12 A__U11 = A__LENGTH A__U11 = MARK A__U12 = A__LENGTH mark < A__U12 A__U12 = MARK A__LENGTH = MARK mark < MARK mark = a__U11 mark = a__U12 mark = a__length a__U11 = a__U12 a__U11 = a__length a__U12 = a__length ---------------------------------------- (14) Obligation: Innermost TRS: Rules: A__ZEROS -> c A__ZEROS -> c1 A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__U11(z0, z1) -> c3 A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) A__U12(z0, z1) -> c5 A__LENGTH(nil) -> c6 A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) A__LENGTH(z0) -> c8 MARK(zeros) -> c9(A__ZEROS) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(0') -> c14 MARK(tt) -> c15 MARK(s(z0)) -> c16(MARK(z0)) MARK(nil) -> c17 a__zeros -> cons(0', zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0' a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0') -> 0' mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Types: A__ZEROS :: c:c1 c :: c:c1 c1 :: c:c1 A__U11 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> c2:c3 tt :: tt:nil:cons:zeros:U11:U12:length:0':s c2 :: c4:c5 -> c2:c3 A__U12 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> c4:c5 c3 :: c2:c3 c4 :: c6:c7:c8 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c4:c5 A__LENGTH :: tt:nil:cons:zeros:U11:U12:length:0':s -> c6:c7:c8 mark :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s MARK :: tt:nil:cons:zeros:U11:U12:length:0':s -> c9:c10:c11:c12:c13:c14:c15:c16:c17 c5 :: c4:c5 nil :: tt:nil:cons:zeros:U11:U12:length:0':s c6 :: c6:c7:c8 cons :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c7 :: c2:c3 -> c6:c7:c8 c8 :: c6:c7:c8 zeros :: tt:nil:cons:zeros:U11:U12:length:0':s c9 :: c:c1 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 U11 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c10 :: c2:c3 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 U12 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c11 :: c4:c5 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 length :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c12 :: c6:c7:c8 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 c13 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 0' :: tt:nil:cons:zeros:U11:U12:length:0':s c14 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 c15 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 s :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c16 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 c17 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 a__zeros :: tt:nil:cons:zeros:U11:U12:length:0':s a__U11 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s a__U12 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s a__length :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s hole_c:c11_18 :: c:c1 hole_c2:c32_18 :: c2:c3 hole_tt:nil:cons:zeros:U11:U12:length:0':s3_18 :: tt:nil:cons:zeros:U11:U12:length:0':s hole_c4:c54_18 :: c4:c5 hole_c6:c7:c85_18 :: c6:c7:c8 hole_c9:c10:c11:c12:c13:c14:c15:c16:c176_18 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18 :: Nat -> tt:nil:cons:zeros:U11:U12:length:0':s gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18 :: Nat -> c9:c10:c11:c12:c13:c14:c15:c16:c17 Generator Equations: gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(0) <=> tt gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(+(x, 1)) <=> cons(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(x), tt) gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18(0) <=> c9(c) gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18(+(x, 1)) <=> c10(c2(c4(c6, c9(c))), gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18(x)) The following defined symbols remain to be analysed: a__U11, A__U11, A__U12, A__LENGTH, mark, MARK, a__U12, a__length They will be analysed ascendingly in the following order: A__U11 = A__U12 A__U11 = A__LENGTH A__U11 = MARK A__U12 = A__LENGTH mark < A__U12 A__U12 = MARK A__LENGTH = MARK mark < MARK mark = a__U11 mark = a__U12 mark = a__length a__U11 = a__U12 a__U11 = a__length a__U12 = a__length ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: mark(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(n81548_18)) -> gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(n81548_18), rt in Omega(0) Induction Base: mark(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(0)) ->_R^Omega(0) tt Induction Step: mark(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(+(n81548_18, 1))) ->_R^Omega(0) cons(mark(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(n81548_18)), tt) ->_IH cons(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(c81549_18), tt) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (16) Obligation: Innermost TRS: Rules: A__ZEROS -> c A__ZEROS -> c1 A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__U11(z0, z1) -> c3 A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) A__U12(z0, z1) -> c5 A__LENGTH(nil) -> c6 A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) A__LENGTH(z0) -> c8 MARK(zeros) -> c9(A__ZEROS) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(0') -> c14 MARK(tt) -> c15 MARK(s(z0)) -> c16(MARK(z0)) MARK(nil) -> c17 a__zeros -> cons(0', zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0' a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0') -> 0' mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Types: A__ZEROS :: c:c1 c :: c:c1 c1 :: c:c1 A__U11 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> c2:c3 tt :: tt:nil:cons:zeros:U11:U12:length:0':s c2 :: c4:c5 -> c2:c3 A__U12 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> c4:c5 c3 :: c2:c3 c4 :: c6:c7:c8 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c4:c5 A__LENGTH :: tt:nil:cons:zeros:U11:U12:length:0':s -> c6:c7:c8 mark :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s MARK :: tt:nil:cons:zeros:U11:U12:length:0':s -> c9:c10:c11:c12:c13:c14:c15:c16:c17 c5 :: c4:c5 nil :: tt:nil:cons:zeros:U11:U12:length:0':s c6 :: c6:c7:c8 cons :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c7 :: c2:c3 -> c6:c7:c8 c8 :: c6:c7:c8 zeros :: tt:nil:cons:zeros:U11:U12:length:0':s c9 :: c:c1 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 U11 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c10 :: c2:c3 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 U12 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c11 :: c4:c5 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 length :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c12 :: c6:c7:c8 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 c13 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 0' :: tt:nil:cons:zeros:U11:U12:length:0':s c14 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 c15 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 s :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c16 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 c17 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 a__zeros :: tt:nil:cons:zeros:U11:U12:length:0':s a__U11 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s a__U12 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s a__length :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s hole_c:c11_18 :: c:c1 hole_c2:c32_18 :: c2:c3 hole_tt:nil:cons:zeros:U11:U12:length:0':s3_18 :: tt:nil:cons:zeros:U11:U12:length:0':s hole_c4:c54_18 :: c4:c5 hole_c6:c7:c85_18 :: c6:c7:c8 hole_c9:c10:c11:c12:c13:c14:c15:c16:c176_18 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18 :: Nat -> tt:nil:cons:zeros:U11:U12:length:0':s gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18 :: Nat -> c9:c10:c11:c12:c13:c14:c15:c16:c17 Lemmas: mark(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(n81548_18)) -> gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(n81548_18), rt in Omega(0) Generator Equations: gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(0) <=> tt gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(+(x, 1)) <=> cons(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(x), tt) gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18(0) <=> c9(c) gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18(+(x, 1)) <=> c10(c2(c4(c6, c9(c))), gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18(x)) The following defined symbols remain to be analysed: a__U11, A__U11, A__U12, A__LENGTH, MARK, a__U12, a__length They will be analysed ascendingly in the following order: A__U11 = A__U12 A__U11 = A__LENGTH A__U11 = MARK A__U12 = A__LENGTH mark < A__U12 A__U12 = MARK A__LENGTH = MARK mark < MARK mark = a__U11 mark = a__U12 mark = a__length a__U11 = a__U12 a__U11 = a__length a__U12 = a__length ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: MARK(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(+(1, n133691_18))) -> *9_18, rt in Omega(n133691_18) Induction Base: MARK(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(+(1, 0))) Induction Step: MARK(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(+(1, +(n133691_18, 1)))) ->_R^Omega(1) c13(MARK(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(+(1, n133691_18)))) ->_IH c13(*9_18) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Complex Obligation (BEST) ---------------------------------------- (19) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: A__ZEROS -> c A__ZEROS -> c1 A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__U11(z0, z1) -> c3 A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) A__U12(z0, z1) -> c5 A__LENGTH(nil) -> c6 A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) A__LENGTH(z0) -> c8 MARK(zeros) -> c9(A__ZEROS) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(0') -> c14 MARK(tt) -> c15 MARK(s(z0)) -> c16(MARK(z0)) MARK(nil) -> c17 a__zeros -> cons(0', zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0' a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0') -> 0' mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Types: A__ZEROS :: c:c1 c :: c:c1 c1 :: c:c1 A__U11 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> c2:c3 tt :: tt:nil:cons:zeros:U11:U12:length:0':s c2 :: c4:c5 -> c2:c3 A__U12 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> c4:c5 c3 :: c2:c3 c4 :: c6:c7:c8 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c4:c5 A__LENGTH :: tt:nil:cons:zeros:U11:U12:length:0':s -> c6:c7:c8 mark :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s MARK :: tt:nil:cons:zeros:U11:U12:length:0':s -> c9:c10:c11:c12:c13:c14:c15:c16:c17 c5 :: c4:c5 nil :: tt:nil:cons:zeros:U11:U12:length:0':s c6 :: c6:c7:c8 cons :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c7 :: c2:c3 -> c6:c7:c8 c8 :: c6:c7:c8 zeros :: tt:nil:cons:zeros:U11:U12:length:0':s c9 :: c:c1 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 U11 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c10 :: c2:c3 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 U12 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c11 :: c4:c5 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 length :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c12 :: c6:c7:c8 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 c13 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 0' :: tt:nil:cons:zeros:U11:U12:length:0':s c14 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 c15 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 s :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c16 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 c17 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 a__zeros :: tt:nil:cons:zeros:U11:U12:length:0':s a__U11 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s a__U12 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s a__length :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s hole_c:c11_18 :: c:c1 hole_c2:c32_18 :: c2:c3 hole_tt:nil:cons:zeros:U11:U12:length:0':s3_18 :: tt:nil:cons:zeros:U11:U12:length:0':s hole_c4:c54_18 :: c4:c5 hole_c6:c7:c85_18 :: c6:c7:c8 hole_c9:c10:c11:c12:c13:c14:c15:c16:c176_18 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18 :: Nat -> tt:nil:cons:zeros:U11:U12:length:0':s gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18 :: Nat -> c9:c10:c11:c12:c13:c14:c15:c16:c17 Lemmas: mark(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(n81548_18)) -> gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(n81548_18), rt in Omega(0) Generator Equations: gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(0) <=> tt gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(+(x, 1)) <=> cons(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(x), tt) gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18(0) <=> c9(c) gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18(+(x, 1)) <=> c10(c2(c4(c6, c9(c))), gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18(x)) The following defined symbols remain to be analysed: MARK They will be analysed ascendingly in the following order: A__U11 = A__U12 A__U11 = A__LENGTH A__U11 = MARK A__U12 = A__LENGTH A__U12 = MARK A__LENGTH = MARK ---------------------------------------- (20) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (21) BOUNDS(n^1, INF) ---------------------------------------- (22) Obligation: Innermost TRS: Rules: A__ZEROS -> c A__ZEROS -> c1 A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__U11(z0, z1) -> c3 A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) A__U12(z0, z1) -> c5 A__LENGTH(nil) -> c6 A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) A__LENGTH(z0) -> c8 MARK(zeros) -> c9(A__ZEROS) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(0') -> c14 MARK(tt) -> c15 MARK(s(z0)) -> c16(MARK(z0)) MARK(nil) -> c17 a__zeros -> cons(0', zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0' a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0') -> 0' mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Types: A__ZEROS :: c:c1 c :: c:c1 c1 :: c:c1 A__U11 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> c2:c3 tt :: tt:nil:cons:zeros:U11:U12:length:0':s c2 :: c4:c5 -> c2:c3 A__U12 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> c4:c5 c3 :: c2:c3 c4 :: c6:c7:c8 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c4:c5 A__LENGTH :: tt:nil:cons:zeros:U11:U12:length:0':s -> c6:c7:c8 mark :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s MARK :: tt:nil:cons:zeros:U11:U12:length:0':s -> c9:c10:c11:c12:c13:c14:c15:c16:c17 c5 :: c4:c5 nil :: tt:nil:cons:zeros:U11:U12:length:0':s c6 :: c6:c7:c8 cons :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c7 :: c2:c3 -> c6:c7:c8 c8 :: c6:c7:c8 zeros :: tt:nil:cons:zeros:U11:U12:length:0':s c9 :: c:c1 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 U11 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c10 :: c2:c3 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 U12 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c11 :: c4:c5 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 length :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c12 :: c6:c7:c8 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 c13 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 0' :: tt:nil:cons:zeros:U11:U12:length:0':s c14 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 c15 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 s :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s c16 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 -> c9:c10:c11:c12:c13:c14:c15:c16:c17 c17 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 a__zeros :: tt:nil:cons:zeros:U11:U12:length:0':s a__U11 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s a__U12 :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s a__length :: tt:nil:cons:zeros:U11:U12:length:0':s -> tt:nil:cons:zeros:U11:U12:length:0':s hole_c:c11_18 :: c:c1 hole_c2:c32_18 :: c2:c3 hole_tt:nil:cons:zeros:U11:U12:length:0':s3_18 :: tt:nil:cons:zeros:U11:U12:length:0':s hole_c4:c54_18 :: c4:c5 hole_c6:c7:c85_18 :: c6:c7:c8 hole_c9:c10:c11:c12:c13:c14:c15:c16:c176_18 :: c9:c10:c11:c12:c13:c14:c15:c16:c17 gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18 :: Nat -> tt:nil:cons:zeros:U11:U12:length:0':s gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18 :: Nat -> c9:c10:c11:c12:c13:c14:c15:c16:c17 Lemmas: mark(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(n81548_18)) -> gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(n81548_18), rt in Omega(0) MARK(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(+(1, n133691_18))) -> *9_18, rt in Omega(n133691_18) Generator Equations: gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(0) <=> tt gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(+(x, 1)) <=> cons(gen_tt:nil:cons:zeros:U11:U12:length:0':s7_18(x), tt) gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18(0) <=> c9(c) gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18(+(x, 1)) <=> c10(c2(c4(c6, c9(c))), gen_c9:c10:c11:c12:c13:c14:c15:c16:c178_18(x)) The following defined symbols remain to be analysed: A__U12, A__U11, A__LENGTH They will be analysed ascendingly in the following order: A__U11 = A__U12 A__U11 = A__LENGTH A__U11 = MARK A__U12 = A__LENGTH A__U12 = MARK A__LENGTH = MARK ---------------------------------------- (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) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__U11(z0, z1) -> c3 A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) A__U12(z0, z1) -> c5 A__LENGTH(nil) -> c6 A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) A__LENGTH(z0) -> c8 MARK(zeros) -> c9(A__ZEROS) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(0) -> c14 MARK(tt) -> c15 MARK(s(z0)) -> c16(MARK(z0)) MARK(nil) -> c17 S tuples: A__ZEROS -> c A__ZEROS -> c1 A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__U11(z0, z1) -> c3 A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) A__U12(z0, z1) -> c5 A__LENGTH(nil) -> c6 A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) A__LENGTH(z0) -> c8 MARK(zeros) -> c9(A__ZEROS) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(0) -> c14 MARK(tt) -> c15 MARK(s(z0)) -> c16(MARK(z0)) MARK(nil) -> c17 K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__ZEROS, A__U11_2, A__U12_2, A__LENGTH_1, MARK_1 Compound Symbols: c, c1, c2_1, c3, c4_2, c5, c6, c7_1, c8, c9_1, c10_2, c11_2, c12_2, c13_1, c14, c15, c16_1, c17 ---------------------------------------- (25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: A__ZEROS -> c1 MARK(tt) -> c15 A__LENGTH(z0) -> c8 A__LENGTH(nil) -> c6 MARK(0) -> c14 MARK(nil) -> c17 MARK(zeros) -> c9(A__ZEROS) A__U12(z0, z1) -> c5 A__ZEROS -> c A__U11(z0, z1) -> c3 ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__U12_2, A__LENGTH_1, MARK_1 Compound Symbols: c2_1, c4_2, c7_1, c10_2, c11_2, c12_2, c13_1, c16_1 ---------------------------------------- (27) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (28) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) The (relative) TRS S consists of the following rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil Rewrite Strategy: INNERMOST ---------------------------------------- (29) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (30) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(tt) -> tt mark(s(z0)) -> s(mark(z0)) mark(nil) -> nil S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (31) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (32) 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__U11(tt, z0) -> c2(A__U12(tt, z0)) [1] A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) [1] A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) [1] MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) [1] MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) [1] MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) [1] MARK(cons(z0, z1)) -> c13(MARK(z0)) [1] MARK(s(z0)) -> c16(MARK(z0)) [1] a__zeros -> cons(0, zeros) [0] a__zeros -> zeros [0] a__U11(tt, z0) -> a__U12(tt, z0) [0] a__U11(z0, z1) -> U11(z0, z1) [0] a__U12(tt, z0) -> s(a__length(mark(z0))) [0] a__U12(z0, z1) -> U12(z0, z1) [0] a__length(nil) -> 0 [0] a__length(cons(z0, z1)) -> a__U11(tt, z1) [0] a__length(z0) -> length(z0) [0] mark(zeros) -> a__zeros [0] mark(U11(z0, z1)) -> a__U11(mark(z0), z1) [0] mark(U12(z0, z1)) -> a__U12(mark(z0), z1) [0] mark(length(z0)) -> a__length(mark(z0)) [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(0) -> 0 [0] mark(tt) -> tt [0] mark(s(z0)) -> s(mark(z0)) [0] mark(nil) -> nil [0] Rewrite Strategy: INNERMOST ---------------------------------------- (33) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (34) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: A__U11(tt, z0) -> c2(A__U12(tt, z0)) [1] A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) [1] A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) [1] MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) [1] MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) [1] MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) [1] MARK(cons(z0, z1)) -> c13(MARK(z0)) [1] MARK(s(z0)) -> c16(MARK(z0)) [1] a__zeros -> cons(0, zeros) [0] a__zeros -> zeros [0] a__U11(tt, z0) -> a__U12(tt, z0) [0] a__U11(z0, z1) -> U11(z0, z1) [0] a__U12(tt, z0) -> s(a__length(mark(z0))) [0] a__U12(z0, z1) -> U12(z0, z1) [0] a__length(nil) -> 0 [0] a__length(cons(z0, z1)) -> a__U11(tt, z1) [0] a__length(z0) -> length(z0) [0] mark(zeros) -> a__zeros [0] mark(U11(z0, z1)) -> a__U11(mark(z0), z1) [0] mark(U12(z0, z1)) -> a__U12(mark(z0), z1) [0] mark(length(z0)) -> a__length(mark(z0)) [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(0) -> 0 [0] mark(tt) -> tt [0] mark(s(z0)) -> s(mark(z0)) [0] mark(nil) -> nil [0] The TRS has the following type information: A__U11 :: tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil -> c2 tt :: tt:cons:U11:U12:length:s:0:zeros:nil c2 :: c4 -> c2 A__U12 :: tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil -> c4 c4 :: c7 -> c10:c11:c12:c13:c16 -> c4 A__LENGTH :: tt:cons:U11:U12:length:s:0:zeros:nil -> c7 mark :: tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil MARK :: tt:cons:U11:U12:length:s:0:zeros:nil -> c10:c11:c12:c13:c16 cons :: tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil c7 :: c2 -> c7 U11 :: tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil c10 :: c2 -> c10:c11:c12:c13:c16 -> c10:c11:c12:c13:c16 U12 :: tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil c11 :: c4 -> c10:c11:c12:c13:c16 -> c10:c11:c12:c13:c16 length :: tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil c12 :: c7 -> c10:c11:c12:c13:c16 -> c10:c11:c12:c13:c16 c13 :: c10:c11:c12:c13:c16 -> c10:c11:c12:c13:c16 s :: tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil c16 :: c10:c11:c12:c13:c16 -> c10:c11:c12:c13:c16 a__zeros :: tt:cons:U11:U12:length:s:0:zeros:nil 0 :: tt:cons:U11:U12:length:s:0:zeros:nil zeros :: tt:cons:U11:U12:length:s:0:zeros:nil a__U11 :: tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil a__U12 :: tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil a__length :: tt:cons:U11:U12:length:s:0:zeros:nil -> tt:cons:U11:U12:length:s:0:zeros:nil nil :: tt:cons:U11:U12:length:s:0:zeros:nil Rewrite Strategy: INNERMOST ---------------------------------------- (35) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: A__U11_2 A__U12_2 A__LENGTH_1 MARK_1 (c) The following functions are completely defined: a__zeros a__U11_2 a__U12_2 a__length_1 mark_1 Due to the following rules being added: a__zeros -> null_a__zeros [0] a__U11(v0, v1) -> null_a__U11 [0] a__U12(v0, v1) -> null_a__U12 [0] a__length(v0) -> null_a__length [0] mark(v0) -> null_mark [0] And the following fresh constants: null_a__zeros, null_a__U11, null_a__U12, null_a__length, null_mark, const, const1, const2, const3 ---------------------------------------- (36) 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__U11(tt, z0) -> c2(A__U12(tt, z0)) [1] A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) [1] A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) [1] MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) [1] MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) [1] MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) [1] MARK(cons(z0, z1)) -> c13(MARK(z0)) [1] MARK(s(z0)) -> c16(MARK(z0)) [1] a__zeros -> cons(0, zeros) [0] a__zeros -> zeros [0] a__U11(tt, z0) -> a__U12(tt, z0) [0] a__U11(z0, z1) -> U11(z0, z1) [0] a__U12(tt, z0) -> s(a__length(mark(z0))) [0] a__U12(z0, z1) -> U12(z0, z1) [0] a__length(nil) -> 0 [0] a__length(cons(z0, z1)) -> a__U11(tt, z1) [0] a__length(z0) -> length(z0) [0] mark(zeros) -> a__zeros [0] mark(U11(z0, z1)) -> a__U11(mark(z0), z1) [0] mark(U12(z0, z1)) -> a__U12(mark(z0), z1) [0] mark(length(z0)) -> a__length(mark(z0)) [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(0) -> 0 [0] mark(tt) -> tt [0] mark(s(z0)) -> s(mark(z0)) [0] mark(nil) -> nil [0] a__zeros -> null_a__zeros [0] a__U11(v0, v1) -> null_a__U11 [0] a__U12(v0, v1) -> null_a__U12 [0] a__length(v0) -> null_a__length [0] mark(v0) -> null_mark [0] The TRS has the following type information: A__U11 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> c2 tt :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c2 :: c4 -> c2 A__U12 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> c4 c4 :: c7 -> c10:c11:c12:c13:c16 -> c4 A__LENGTH :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> c7 mark :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark MARK :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> c10:c11:c12:c13:c16 cons :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c7 :: c2 -> c7 U11 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c10 :: c2 -> c10:c11:c12:c13:c16 -> c10:c11:c12:c13:c16 U12 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c11 :: c4 -> c10:c11:c12:c13:c16 -> c10:c11:c12:c13:c16 length :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c12 :: c7 -> c10:c11:c12:c13:c16 -> c10:c11:c12:c13:c16 c13 :: c10:c11:c12:c13:c16 -> c10:c11:c12:c13:c16 s :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c16 :: c10:c11:c12:c13:c16 -> c10:c11:c12:c13:c16 a__zeros :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark 0 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark zeros :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark a__U11 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark a__U12 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark a__length :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark nil :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_a__zeros :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_a__U11 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_a__U12 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_a__length :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_mark :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark const :: c2 const1 :: c4 const2 :: c7 const3 :: c10:c11:c12:c13:c16 Rewrite Strategy: INNERMOST ---------------------------------------- (37) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (38) 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__U11(tt, z0) -> c2(A__U12(tt, z0)) [1] A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros), MARK(zeros)) [1] A__U12(tt, U11(z0', z1')) -> c4(A__LENGTH(a__U11(mark(z0'), z1')), MARK(U11(z0', z1'))) [1] A__U12(tt, U12(z0'', z1'')) -> c4(A__LENGTH(a__U12(mark(z0''), z1'')), MARK(U12(z0'', z1''))) [1] A__U12(tt, length(z01)) -> c4(A__LENGTH(a__length(mark(z01))), MARK(length(z01))) [1] A__U12(tt, cons(z02, z11)) -> c4(A__LENGTH(cons(mark(z02), z11)), MARK(cons(z02, z11))) [1] A__U12(tt, 0) -> c4(A__LENGTH(0), MARK(0)) [1] A__U12(tt, tt) -> c4(A__LENGTH(tt), MARK(tt)) [1] A__U12(tt, s(z03)) -> c4(A__LENGTH(s(mark(z03))), MARK(s(z03))) [1] A__U12(tt, nil) -> c4(A__LENGTH(nil), MARK(nil)) [1] A__U12(tt, z0) -> c4(A__LENGTH(null_mark), MARK(z0)) [1] A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) [1] MARK(U11(zeros, z1)) -> c10(A__U11(a__zeros, z1), MARK(zeros)) [1] MARK(U11(U11(z04, z12), z1)) -> c10(A__U11(a__U11(mark(z04), z12), z1), MARK(U11(z04, z12))) [1] MARK(U11(U12(z05, z13), z1)) -> c10(A__U11(a__U12(mark(z05), z13), z1), MARK(U12(z05, z13))) [1] MARK(U11(length(z06), z1)) -> c10(A__U11(a__length(mark(z06)), z1), MARK(length(z06))) [1] MARK(U11(cons(z07, z14), z1)) -> c10(A__U11(cons(mark(z07), z14), z1), MARK(cons(z07, z14))) [1] MARK(U11(0, z1)) -> c10(A__U11(0, z1), MARK(0)) [1] MARK(U11(tt, z1)) -> c10(A__U11(tt, z1), MARK(tt)) [1] MARK(U11(s(z08), z1)) -> c10(A__U11(s(mark(z08)), z1), MARK(s(z08))) [1] MARK(U11(nil, z1)) -> c10(A__U11(nil, z1), MARK(nil)) [1] MARK(U11(z0, z1)) -> c10(A__U11(null_mark, z1), MARK(z0)) [1] MARK(U12(zeros, z1)) -> c11(A__U12(a__zeros, z1), MARK(zeros)) [1] MARK(U12(U11(z09, z15), z1)) -> c11(A__U12(a__U11(mark(z09), z15), z1), MARK(U11(z09, z15))) [1] MARK(U12(U12(z010, z16), z1)) -> c11(A__U12(a__U12(mark(z010), z16), z1), MARK(U12(z010, z16))) [1] MARK(U12(length(z011), z1)) -> c11(A__U12(a__length(mark(z011)), z1), MARK(length(z011))) [1] MARK(U12(cons(z012, z17), z1)) -> c11(A__U12(cons(mark(z012), z17), z1), MARK(cons(z012, z17))) [1] MARK(U12(0, z1)) -> c11(A__U12(0, z1), MARK(0)) [1] MARK(U12(tt, z1)) -> c11(A__U12(tt, z1), MARK(tt)) [1] MARK(U12(s(z013), z1)) -> c11(A__U12(s(mark(z013)), z1), MARK(s(z013))) [1] MARK(U12(nil, z1)) -> c11(A__U12(nil, z1), MARK(nil)) [1] MARK(U12(z0, z1)) -> c11(A__U12(null_mark, z1), MARK(z0)) [1] MARK(length(zeros)) -> c12(A__LENGTH(a__zeros), MARK(zeros)) [1] MARK(length(U11(z014, z18))) -> c12(A__LENGTH(a__U11(mark(z014), z18)), MARK(U11(z014, z18))) [1] MARK(length(U12(z015, z19))) -> c12(A__LENGTH(a__U12(mark(z015), z19)), MARK(U12(z015, z19))) [1] MARK(length(length(z016))) -> c12(A__LENGTH(a__length(mark(z016))), MARK(length(z016))) [1] MARK(length(cons(z017, z110))) -> c12(A__LENGTH(cons(mark(z017), z110)), MARK(cons(z017, z110))) [1] MARK(length(0)) -> c12(A__LENGTH(0), MARK(0)) [1] MARK(length(tt)) -> c12(A__LENGTH(tt), MARK(tt)) [1] MARK(length(s(z018))) -> c12(A__LENGTH(s(mark(z018))), MARK(s(z018))) [1] MARK(length(nil)) -> c12(A__LENGTH(nil), MARK(nil)) [1] MARK(length(z0)) -> c12(A__LENGTH(null_mark), MARK(z0)) [1] MARK(cons(z0, z1)) -> c13(MARK(z0)) [1] MARK(s(z0)) -> c16(MARK(z0)) [1] a__zeros -> cons(0, zeros) [0] a__zeros -> zeros [0] a__U11(tt, z0) -> a__U12(tt, z0) [0] a__U11(z0, z1) -> U11(z0, z1) [0] a__U12(tt, zeros) -> s(a__length(a__zeros)) [0] a__U12(tt, U11(z019, z111)) -> s(a__length(a__U11(mark(z019), z111))) [0] a__U12(tt, U12(z020, z112)) -> s(a__length(a__U12(mark(z020), z112))) [0] a__U12(tt, length(z021)) -> s(a__length(a__length(mark(z021)))) [0] a__U12(tt, cons(z022, z113)) -> s(a__length(cons(mark(z022), z113))) [0] a__U12(tt, 0) -> s(a__length(0)) [0] a__U12(tt, tt) -> s(a__length(tt)) [0] a__U12(tt, s(z023)) -> s(a__length(s(mark(z023)))) [0] a__U12(tt, nil) -> s(a__length(nil)) [0] a__U12(tt, z0) -> s(a__length(null_mark)) [0] a__U12(z0, z1) -> U12(z0, z1) [0] a__length(nil) -> 0 [0] a__length(cons(z0, z1)) -> a__U11(tt, z1) [0] a__length(z0) -> length(z0) [0] mark(zeros) -> a__zeros [0] mark(U11(zeros, z1)) -> a__U11(a__zeros, z1) [0] mark(U11(U11(z024, z114), z1)) -> a__U11(a__U11(mark(z024), z114), z1) [0] mark(U11(U12(z025, z115), z1)) -> a__U11(a__U12(mark(z025), z115), z1) [0] mark(U11(length(z026), z1)) -> a__U11(a__length(mark(z026)), z1) [0] mark(U11(cons(z027, z116), z1)) -> a__U11(cons(mark(z027), z116), z1) [0] mark(U11(0, z1)) -> a__U11(0, z1) [0] mark(U11(tt, z1)) -> a__U11(tt, z1) [0] mark(U11(s(z028), z1)) -> a__U11(s(mark(z028)), z1) [0] mark(U11(nil, z1)) -> a__U11(nil, z1) [0] mark(U11(z0, z1)) -> a__U11(null_mark, z1) [0] mark(U12(zeros, z1)) -> a__U12(a__zeros, z1) [0] mark(U12(U11(z029, z117), z1)) -> a__U12(a__U11(mark(z029), z117), z1) [0] mark(U12(U12(z030, z118), z1)) -> a__U12(a__U12(mark(z030), z118), z1) [0] mark(U12(length(z031), z1)) -> a__U12(a__length(mark(z031)), z1) [0] mark(U12(cons(z032, z119), z1)) -> a__U12(cons(mark(z032), z119), z1) [0] mark(U12(0, z1)) -> a__U12(0, z1) [0] mark(U12(tt, z1)) -> a__U12(tt, z1) [0] mark(U12(s(z033), z1)) -> a__U12(s(mark(z033)), z1) [0] mark(U12(nil, z1)) -> a__U12(nil, z1) [0] mark(U12(z0, z1)) -> a__U12(null_mark, z1) [0] mark(length(zeros)) -> a__length(a__zeros) [0] mark(length(U11(z034, z120))) -> a__length(a__U11(mark(z034), z120)) [0] mark(length(U12(z035, z121))) -> a__length(a__U12(mark(z035), z121)) [0] mark(length(length(z036))) -> a__length(a__length(mark(z036))) [0] mark(length(cons(z037, z122))) -> a__length(cons(mark(z037), z122)) [0] mark(length(0)) -> a__length(0) [0] mark(length(tt)) -> a__length(tt) [0] mark(length(s(z038))) -> a__length(s(mark(z038))) [0] mark(length(nil)) -> a__length(nil) [0] mark(length(z0)) -> a__length(null_mark) [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(0) -> 0 [0] mark(tt) -> tt [0] mark(s(z0)) -> s(mark(z0)) [0] mark(nil) -> nil [0] a__zeros -> null_a__zeros [0] a__U11(v0, v1) -> null_a__U11 [0] a__U12(v0, v1) -> null_a__U12 [0] a__length(v0) -> null_a__length [0] mark(v0) -> null_mark [0] The TRS has the following type information: A__U11 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> c2 tt :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c2 :: c4 -> c2 A__U12 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> c4 c4 :: c7 -> c10:c11:c12:c13:c16 -> c4 A__LENGTH :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> c7 mark :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark MARK :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> c10:c11:c12:c13:c16 cons :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c7 :: c2 -> c7 U11 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c10 :: c2 -> c10:c11:c12:c13:c16 -> c10:c11:c12:c13:c16 U12 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c11 :: c4 -> c10:c11:c12:c13:c16 -> c10:c11:c12:c13:c16 length :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c12 :: c7 -> c10:c11:c12:c13:c16 -> c10:c11:c12:c13:c16 c13 :: c10:c11:c12:c13:c16 -> c10:c11:c12:c13:c16 s :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c16 :: c10:c11:c12:c13:c16 -> c10:c11:c12:c13:c16 a__zeros :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark 0 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark zeros :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark a__U11 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark a__U12 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark a__length :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark nil :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_a__zeros :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_a__U11 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_a__U12 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_a__length :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_mark :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark const :: c2 const1 :: c4 const2 :: c7 const3 :: c10:c11:c12:c13:c16 Rewrite Strategy: INNERMOST ---------------------------------------- (39) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: tt => 2 0 => 0 zeros => 3 nil => 1 null_a__zeros => 0 null_a__U11 => 0 null_a__U12 => 0 null_a__length => 0 null_mark => 0 const => 0 const1 => 0 const2 => 0 const3 => 0 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: A__LENGTH(z) -{ 1 }-> 1 + A__U11(2, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__U11(z, z') -{ 1 }-> 1 + A__U12(2, z0) :|: z = 2, z0 >= 0, z' = z0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(a__zeros) + MARK(3) :|: z = 2, z' = 3 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(a__length(mark(z01))) + MARK(1 + z01) :|: z = 2, z01 >= 0, z' = 1 + z01 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(a__U12(mark(z0''), z1'')) + MARK(1 + z0'' + z1'') :|: z = 2, z0'' >= 0, z' = 1 + z0'' + z1'', z1'' >= 0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(a__U11(mark(z0'), z1')) + MARK(1 + z0' + z1') :|: z = 2, z0' >= 0, z1' >= 0, z' = 1 + z0' + z1' A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(2) + MARK(2) :|: z = 2, z' = 2 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(1) + MARK(1) :|: z = 2, z' = 1 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(0) + MARK(z0) :|: z = 2, z0 >= 0, z' = z0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(0) + MARK(0) :|: z = 2, z' = 0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(1 + mark(z03)) + MARK(1 + z03) :|: z = 2, z' = 1 + z03, z03 >= 0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(1 + mark(z02) + z11) + MARK(1 + z02 + z11) :|: z = 2, z11 >= 0, z02 >= 0, z' = 1 + z02 + z11 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(a__zeros, z1) + MARK(3) :|: z1 >= 0, z = 1 + 3 + z1 MARK(z) -{ 1 }-> 1 + A__U12(a__length(mark(z011)), z1) + MARK(1 + z011) :|: z011 >= 0, z1 >= 0, z = 1 + (1 + z011) + z1 MARK(z) -{ 1 }-> 1 + A__U12(a__U12(mark(z010), z16), z1) + MARK(1 + z010 + z16) :|: z1 >= 0, z = 1 + (1 + z010 + z16) + z1, z16 >= 0, z010 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(a__U11(mark(z09), z15), z1) + MARK(1 + z09 + z15) :|: z15 >= 0, z1 >= 0, z = 1 + (1 + z09 + z15) + z1, z09 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(2, z1) + MARK(2) :|: z1 >= 0, z = 1 + 2 + z1 MARK(z) -{ 1 }-> 1 + A__U12(1, z1) + MARK(1) :|: z1 >= 0, z = 1 + 1 + z1 MARK(z) -{ 1 }-> 1 + A__U12(0, z1) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__U12(0, z1) + MARK(0) :|: z1 >= 0, z = 1 + 0 + z1 MARK(z) -{ 1 }-> 1 + A__U12(1 + mark(z013), z1) + MARK(1 + z013) :|: z1 >= 0, z013 >= 0, z = 1 + (1 + z013) + z1 MARK(z) -{ 1 }-> 1 + A__U12(1 + mark(z012) + z17, z1) + MARK(1 + z012 + z17) :|: z1 >= 0, z = 1 + (1 + z012 + z17) + z1, z17 >= 0, z012 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(a__zeros, z1) + MARK(3) :|: z1 >= 0, z = 1 + 3 + z1 MARK(z) -{ 1 }-> 1 + A__U11(a__length(mark(z06)), z1) + MARK(1 + z06) :|: z1 >= 0, z = 1 + (1 + z06) + z1, z06 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(a__U12(mark(z05), z13), z1) + MARK(1 + z05 + z13) :|: z1 >= 0, z05 >= 0, z = 1 + (1 + z05 + z13) + z1, z13 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(a__U11(mark(z04), z12), z1) + MARK(1 + z04 + z12) :|: z04 >= 0, z = 1 + (1 + z04 + z12) + z1, z1 >= 0, z12 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(2, z1) + MARK(2) :|: z1 >= 0, z = 1 + 2 + z1 MARK(z) -{ 1 }-> 1 + A__U11(1, z1) + MARK(1) :|: z1 >= 0, z = 1 + 1 + z1 MARK(z) -{ 1 }-> 1 + A__U11(0, z1) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__U11(0, z1) + MARK(0) :|: z1 >= 0, z = 1 + 0 + z1 MARK(z) -{ 1 }-> 1 + A__U11(1 + mark(z08), z1) + MARK(1 + z08) :|: z08 >= 0, z1 >= 0, z = 1 + (1 + z08) + z1 MARK(z) -{ 1 }-> 1 + A__U11(1 + mark(z07) + z14, z1) + MARK(1 + z07 + z14) :|: z1 >= 0, z07 >= 0, z = 1 + (1 + z07 + z14) + z1, z14 >= 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(a__zeros) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__LENGTH(a__length(mark(z016))) + MARK(1 + z016) :|: z016 >= 0, z = 1 + (1 + z016) MARK(z) -{ 1 }-> 1 + A__LENGTH(a__U12(mark(z015), z19)) + MARK(1 + z015 + z19) :|: z015 >= 0, z19 >= 0, z = 1 + (1 + z015 + z19) MARK(z) -{ 1 }-> 1 + A__LENGTH(a__U11(mark(z014), z18)) + MARK(1 + z014 + z18) :|: z = 1 + (1 + z014 + z18), z18 >= 0, z014 >= 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(2) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__LENGTH(1) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__LENGTH(0) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(1 + mark(z018)) + MARK(1 + z018) :|: z = 1 + (1 + z018), z018 >= 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(1 + mark(z017) + z110) + MARK(1 + z017 + z110) :|: z110 >= 0, z = 1 + (1 + z017 + z110), z017 >= 0 a__U11(z, z') -{ 0 }-> a__U12(2, z0) :|: z = 2, z0 >= 0, z' = z0 a__U11(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a__U11(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 a__U12(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a__U12(z, z') -{ 0 }-> 1 + a__length(a__zeros) :|: z = 2, z' = 3 a__U12(z, z') -{ 0 }-> 1 + a__length(a__length(mark(z021))) :|: z = 2, z021 >= 0, z' = 1 + z021 a__U12(z, z') -{ 0 }-> 1 + a__length(a__U12(mark(z020), z112)) :|: z = 2, z' = 1 + z020 + z112, z112 >= 0, z020 >= 0 a__U12(z, z') -{ 0 }-> 1 + a__length(a__U11(mark(z019), z111)) :|: z = 2, z019 >= 0, z' = 1 + z019 + z111, z111 >= 0 a__U12(z, z') -{ 0 }-> 1 + a__length(2) :|: z = 2, z' = 2 a__U12(z, z') -{ 0 }-> 1 + a__length(1) :|: z = 2, z' = 1 a__U12(z, z') -{ 0 }-> 1 + a__length(0) :|: z = 2, z' = 0 a__U12(z, z') -{ 0 }-> 1 + a__length(0) :|: z = 2, z0 >= 0, z' = z0 a__U12(z, z') -{ 0 }-> 1 + a__length(1 + mark(z023)) :|: z = 2, z' = 1 + z023, z023 >= 0 a__U12(z, z') -{ 0 }-> 1 + a__length(1 + mark(z022) + z113) :|: z = 2, z022 >= 0, z113 >= 0, z' = 1 + z022 + z113 a__U12(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 a__length(z) -{ 0 }-> a__U11(2, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__length(z) -{ 0 }-> 0 :|: z = 1 a__length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__length(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__zeros -{ 0 }-> 3 :|: a__zeros -{ 0 }-> 0 :|: a__zeros -{ 0 }-> 1 + 0 + 3 :|: mark(z) -{ 0 }-> a__zeros :|: z = 3 mark(z) -{ 0 }-> a__length(a__zeros) :|: z = 1 + 3 mark(z) -{ 0 }-> a__length(a__length(mark(z036))) :|: z036 >= 0, z = 1 + (1 + z036) mark(z) -{ 0 }-> a__length(a__U12(mark(z035), z121)) :|: z035 >= 0, z = 1 + (1 + z035 + z121), z121 >= 0 mark(z) -{ 0 }-> a__length(a__U11(mark(z034), z120)) :|: z120 >= 0, z = 1 + (1 + z034 + z120), z034 >= 0 mark(z) -{ 0 }-> a__length(2) :|: z = 1 + 2 mark(z) -{ 0 }-> a__length(1) :|: z = 1 + 1 mark(z) -{ 0 }-> a__length(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__length(0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__length(1 + mark(z038)) :|: z = 1 + (1 + z038), z038 >= 0 mark(z) -{ 0 }-> a__length(1 + mark(z037) + z122) :|: z = 1 + (1 + z037 + z122), z122 >= 0, z037 >= 0 mark(z) -{ 0 }-> a__U12(a__zeros, z1) :|: z1 >= 0, z = 1 + 3 + z1 mark(z) -{ 0 }-> a__U12(a__length(mark(z031)), z1) :|: z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0 mark(z) -{ 0 }-> a__U12(a__U12(mark(z030), z118), z1) :|: z1 >= 0, z118 >= 0, z = 1 + (1 + z030 + z118) + z1, z030 >= 0 mark(z) -{ 0 }-> a__U12(a__U11(mark(z029), z117), z1) :|: z029 >= 0, z117 >= 0, z1 >= 0, z = 1 + (1 + z029 + z117) + z1 mark(z) -{ 0 }-> a__U12(2, z1) :|: z1 >= 0, z = 1 + 2 + z1 mark(z) -{ 0 }-> a__U12(1, z1) :|: z1 >= 0, z = 1 + 1 + z1 mark(z) -{ 0 }-> a__U12(0, z1) :|: z1 >= 0, z = 1 + 0 + z1 mark(z) -{ 0 }-> a__U12(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> a__U12(1 + mark(z033), z1) :|: z1 >= 0, z = 1 + (1 + z033) + z1, z033 >= 0 mark(z) -{ 0 }-> a__U12(1 + mark(z032) + z119, z1) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032 + z119) + z1, z119 >= 0 mark(z) -{ 0 }-> a__U11(a__zeros, z1) :|: z1 >= 0, z = 1 + 3 + z1 mark(z) -{ 0 }-> a__U11(a__length(mark(z026)), z1) :|: z1 >= 0, z = 1 + (1 + z026) + z1, z026 >= 0 mark(z) -{ 0 }-> a__U11(a__U12(mark(z025), z115), z1) :|: z025 >= 0, z1 >= 0, z115 >= 0, z = 1 + (1 + z025 + z115) + z1 mark(z) -{ 0 }-> a__U11(a__U11(mark(z024), z114), z1) :|: z114 >= 0, z1 >= 0, z024 >= 0, z = 1 + (1 + z024 + z114) + z1 mark(z) -{ 0 }-> a__U11(2, z1) :|: z1 >= 0, z = 1 + 2 + z1 mark(z) -{ 0 }-> a__U11(1, z1) :|: z1 >= 0, z = 1 + 1 + z1 mark(z) -{ 0 }-> a__U11(0, z1) :|: z1 >= 0, z = 1 + 0 + z1 mark(z) -{ 0 }-> a__U11(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> a__U11(1 + mark(z028), z1) :|: z1 >= 0, z = 1 + (1 + z028) + z1, z028 >= 0 mark(z) -{ 0 }-> a__U11(1 + mark(z027) + z116, z1) :|: z1 >= 0, z027 >= 0, z = 1 + (1 + z027 + z116) + z1, z116 >= 0 mark(z) -{ 0 }-> 2 :|: z = 2 mark(z) -{ 0 }-> 1 :|: z = 1 mark(z) -{ 0 }-> 0 :|: z = 0 mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 mark(z) -{ 0 }-> 1 + mark(z0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> 1 + mark(z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 ---------------------------------------- (41) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: a__zeros -{ 0 }-> 3 :|: a__zeros -{ 0 }-> 0 :|: a__zeros -{ 0 }-> 1 + 0 + 3 :|: ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: A__LENGTH(z) -{ 1 }-> 1 + A__U11(2, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__U11(z, z') -{ 1 }-> 1 + A__U12(2, z0) :|: z = 2, z0 >= 0, z' = z0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(a__length(mark(z01))) + MARK(1 + z01) :|: z = 2, z01 >= 0, z' = 1 + z01 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(a__U12(mark(z0''), z1'')) + MARK(1 + z0'' + z1'') :|: z = 2, z0'' >= 0, z' = 1 + z0'' + z1'', z1'' >= 0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(a__U11(mark(z0'), z1')) + MARK(1 + z0' + z1') :|: z = 2, z0' >= 0, z1' >= 0, z' = 1 + z0' + z1' A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(3) + MARK(3) :|: z = 2, z' = 3 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(2) + MARK(2) :|: z = 2, z' = 2 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(1) + MARK(1) :|: z = 2, z' = 1 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(0) + MARK(z0) :|: z = 2, z0 >= 0, z' = z0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(0) + MARK(3) :|: z = 2, z' = 3 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(0) + MARK(0) :|: z = 2, z' = 0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(1 + mark(z03)) + MARK(1 + z03) :|: z = 2, z' = 1 + z03, z03 >= 0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(1 + mark(z02) + z11) + MARK(1 + z02 + z11) :|: z = 2, z11 >= 0, z02 >= 0, z' = 1 + z02 + z11 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(1 + 0 + 3) + MARK(3) :|: z = 2, z' = 3 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(a__length(mark(z011)), z1) + MARK(1 + z011) :|: z011 >= 0, z1 >= 0, z = 1 + (1 + z011) + z1 MARK(z) -{ 1 }-> 1 + A__U12(a__U12(mark(z010), z16), z1) + MARK(1 + z010 + z16) :|: z1 >= 0, z = 1 + (1 + z010 + z16) + z1, z16 >= 0, z010 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(a__U11(mark(z09), z15), z1) + MARK(1 + z09 + z15) :|: z15 >= 0, z1 >= 0, z = 1 + (1 + z09 + z15) + z1, z09 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(3, z1) + MARK(3) :|: z1 >= 0, z = 1 + 3 + z1 MARK(z) -{ 1 }-> 1 + A__U12(2, z1) + MARK(2) :|: z1 >= 0, z = 1 + 2 + z1 MARK(z) -{ 1 }-> 1 + A__U12(1, z1) + MARK(1) :|: z1 >= 0, z = 1 + 1 + z1 MARK(z) -{ 1 }-> 1 + A__U12(0, z1) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__U12(0, z1) + MARK(3) :|: z1 >= 0, z = 1 + 3 + z1 MARK(z) -{ 1 }-> 1 + A__U12(0, z1) + MARK(0) :|: z1 >= 0, z = 1 + 0 + z1 MARK(z) -{ 1 }-> 1 + A__U12(1 + mark(z013), z1) + MARK(1 + z013) :|: z1 >= 0, z013 >= 0, z = 1 + (1 + z013) + z1 MARK(z) -{ 1 }-> 1 + A__U12(1 + mark(z012) + z17, z1) + MARK(1 + z012 + z17) :|: z1 >= 0, z = 1 + (1 + z012 + z17) + z1, z17 >= 0, z012 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(1 + 0 + 3, z1) + MARK(3) :|: z1 >= 0, z = 1 + 3 + z1 MARK(z) -{ 1 }-> 1 + A__U11(a__length(mark(z06)), z1) + MARK(1 + z06) :|: z1 >= 0, z = 1 + (1 + z06) + z1, z06 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(a__U12(mark(z05), z13), z1) + MARK(1 + z05 + z13) :|: z1 >= 0, z05 >= 0, z = 1 + (1 + z05 + z13) + z1, z13 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(a__U11(mark(z04), z12), z1) + MARK(1 + z04 + z12) :|: z04 >= 0, z = 1 + (1 + z04 + z12) + z1, z1 >= 0, z12 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(3, z1) + MARK(3) :|: z1 >= 0, z = 1 + 3 + z1 MARK(z) -{ 1 }-> 1 + A__U11(2, z1) + MARK(2) :|: z1 >= 0, z = 1 + 2 + z1 MARK(z) -{ 1 }-> 1 + A__U11(1, z1) + MARK(1) :|: z1 >= 0, z = 1 + 1 + z1 MARK(z) -{ 1 }-> 1 + A__U11(0, z1) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__U11(0, z1) + MARK(3) :|: z1 >= 0, z = 1 + 3 + z1 MARK(z) -{ 1 }-> 1 + A__U11(0, z1) + MARK(0) :|: z1 >= 0, z = 1 + 0 + z1 MARK(z) -{ 1 }-> 1 + A__U11(1 + mark(z08), z1) + MARK(1 + z08) :|: z08 >= 0, z1 >= 0, z = 1 + (1 + z08) + z1 MARK(z) -{ 1 }-> 1 + A__U11(1 + mark(z07) + z14, z1) + MARK(1 + z07 + z14) :|: z1 >= 0, z07 >= 0, z = 1 + (1 + z07 + z14) + z1, z14 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(1 + 0 + 3, z1) + MARK(3) :|: z1 >= 0, z = 1 + 3 + z1 MARK(z) -{ 1 }-> 1 + A__LENGTH(a__length(mark(z016))) + MARK(1 + z016) :|: z016 >= 0, z = 1 + (1 + z016) MARK(z) -{ 1 }-> 1 + A__LENGTH(a__U12(mark(z015), z19)) + MARK(1 + z015 + z19) :|: z015 >= 0, z19 >= 0, z = 1 + (1 + z015 + z19) MARK(z) -{ 1 }-> 1 + A__LENGTH(a__U11(mark(z014), z18)) + MARK(1 + z014 + z18) :|: z = 1 + (1 + z014 + z18), z18 >= 0, z014 >= 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__LENGTH(2) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__LENGTH(1) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__LENGTH(0) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(0) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__LENGTH(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(1 + mark(z018)) + MARK(1 + z018) :|: z = 1 + (1 + z018), z018 >= 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(1 + mark(z017) + z110) + MARK(1 + z017 + z110) :|: z110 >= 0, z = 1 + (1 + z017 + z110), z017 >= 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(1 + 0 + 3) + MARK(3) :|: z = 1 + 3 a__U11(z, z') -{ 0 }-> a__U12(2, z0) :|: z = 2, z0 >= 0, z' = z0 a__U11(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a__U11(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 a__U12(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a__U12(z, z') -{ 0 }-> 1 + a__length(a__length(mark(z021))) :|: z = 2, z021 >= 0, z' = 1 + z021 a__U12(z, z') -{ 0 }-> 1 + a__length(a__U12(mark(z020), z112)) :|: z = 2, z' = 1 + z020 + z112, z112 >= 0, z020 >= 0 a__U12(z, z') -{ 0 }-> 1 + a__length(a__U11(mark(z019), z111)) :|: z = 2, z019 >= 0, z' = 1 + z019 + z111, z111 >= 0 a__U12(z, z') -{ 0 }-> 1 + a__length(3) :|: z = 2, z' = 3 a__U12(z, z') -{ 0 }-> 1 + a__length(2) :|: z = 2, z' = 2 a__U12(z, z') -{ 0 }-> 1 + a__length(1) :|: z = 2, z' = 1 a__U12(z, z') -{ 0 }-> 1 + a__length(0) :|: z = 2, z' = 0 a__U12(z, z') -{ 0 }-> 1 + a__length(0) :|: z = 2, z0 >= 0, z' = z0 a__U12(z, z') -{ 0 }-> 1 + a__length(0) :|: z = 2, z' = 3 a__U12(z, z') -{ 0 }-> 1 + a__length(1 + mark(z023)) :|: z = 2, z' = 1 + z023, z023 >= 0 a__U12(z, z') -{ 0 }-> 1 + a__length(1 + mark(z022) + z113) :|: z = 2, z022 >= 0, z113 >= 0, z' = 1 + z022 + z113 a__U12(z, z') -{ 0 }-> 1 + a__length(1 + 0 + 3) :|: z = 2, z' = 3 a__U12(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 a__length(z) -{ 0 }-> a__U11(2, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__length(z) -{ 0 }-> 0 :|: z = 1 a__length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__length(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__zeros -{ 0 }-> 3 :|: a__zeros -{ 0 }-> 0 :|: a__zeros -{ 0 }-> 1 + 0 + 3 :|: mark(z) -{ 0 }-> a__length(a__length(mark(z036))) :|: z036 >= 0, z = 1 + (1 + z036) mark(z) -{ 0 }-> a__length(a__U12(mark(z035), z121)) :|: z035 >= 0, z = 1 + (1 + z035 + z121), z121 >= 0 mark(z) -{ 0 }-> a__length(a__U11(mark(z034), z120)) :|: z120 >= 0, z = 1 + (1 + z034 + z120), z034 >= 0 mark(z) -{ 0 }-> a__length(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__length(2) :|: z = 1 + 2 mark(z) -{ 0 }-> a__length(1) :|: z = 1 + 1 mark(z) -{ 0 }-> a__length(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__length(0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__length(0) :|: z = 1 + 3 mark(z) -{ 0 }-> a__length(1 + mark(z038)) :|: z = 1 + (1 + z038), z038 >= 0 mark(z) -{ 0 }-> a__length(1 + mark(z037) + z122) :|: z = 1 + (1 + z037 + z122), z122 >= 0, z037 >= 0 mark(z) -{ 0 }-> a__length(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__U12(a__length(mark(z031)), z1) :|: z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0 mark(z) -{ 0 }-> a__U12(a__U12(mark(z030), z118), z1) :|: z1 >= 0, z118 >= 0, z = 1 + (1 + z030 + z118) + z1, z030 >= 0 mark(z) -{ 0 }-> a__U12(a__U11(mark(z029), z117), z1) :|: z029 >= 0, z117 >= 0, z1 >= 0, z = 1 + (1 + z029 + z117) + z1 mark(z) -{ 0 }-> a__U12(3, z1) :|: z1 >= 0, z = 1 + 3 + z1 mark(z) -{ 0 }-> a__U12(2, z1) :|: z1 >= 0, z = 1 + 2 + z1 mark(z) -{ 0 }-> a__U12(1, z1) :|: z1 >= 0, z = 1 + 1 + z1 mark(z) -{ 0 }-> a__U12(0, z1) :|: z1 >= 0, z = 1 + 0 + z1 mark(z) -{ 0 }-> a__U12(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> a__U12(0, z1) :|: z1 >= 0, z = 1 + 3 + z1 mark(z) -{ 0 }-> a__U12(1 + mark(z033), z1) :|: z1 >= 0, z = 1 + (1 + z033) + z1, z033 >= 0 mark(z) -{ 0 }-> a__U12(1 + mark(z032) + z119, z1) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032 + z119) + z1, z119 >= 0 mark(z) -{ 0 }-> a__U12(1 + 0 + 3, z1) :|: z1 >= 0, z = 1 + 3 + z1 mark(z) -{ 0 }-> a__U11(a__length(mark(z026)), z1) :|: z1 >= 0, z = 1 + (1 + z026) + z1, z026 >= 0 mark(z) -{ 0 }-> a__U11(a__U12(mark(z025), z115), z1) :|: z025 >= 0, z1 >= 0, z115 >= 0, z = 1 + (1 + z025 + z115) + z1 mark(z) -{ 0 }-> a__U11(a__U11(mark(z024), z114), z1) :|: z114 >= 0, z1 >= 0, z024 >= 0, z = 1 + (1 + z024 + z114) + z1 mark(z) -{ 0 }-> a__U11(3, z1) :|: z1 >= 0, z = 1 + 3 + z1 mark(z) -{ 0 }-> a__U11(2, z1) :|: z1 >= 0, z = 1 + 2 + z1 mark(z) -{ 0 }-> a__U11(1, z1) :|: z1 >= 0, z = 1 + 1 + z1 mark(z) -{ 0 }-> a__U11(0, z1) :|: z1 >= 0, z = 1 + 0 + z1 mark(z) -{ 0 }-> a__U11(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> a__U11(0, z1) :|: z1 >= 0, z = 1 + 3 + z1 mark(z) -{ 0 }-> a__U11(1 + mark(z028), z1) :|: z1 >= 0, z = 1 + (1 + z028) + z1, z028 >= 0 mark(z) -{ 0 }-> a__U11(1 + mark(z027) + z116, z1) :|: z1 >= 0, z027 >= 0, z = 1 + (1 + z027 + z116) + z1, z116 >= 0 mark(z) -{ 0 }-> a__U11(1 + 0 + 3, z1) :|: z1 >= 0, z = 1 + 3 + z1 mark(z) -{ 0 }-> 3 :|: z = 3 mark(z) -{ 0 }-> 2 :|: z = 2 mark(z) -{ 0 }-> 1 :|: z = 1 mark(z) -{ 0 }-> 0 :|: z = 0 mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 mark(z) -{ 0 }-> 0 :|: z = 3 mark(z) -{ 0 }-> 1 + mark(z0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> 1 + mark(z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> 1 + 0 + 3 :|: z = 3 ---------------------------------------- (43) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: A__LENGTH(z) -{ 1 }-> 1 + A__U11(2, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__U11(z, z') -{ 1 }-> 1 + A__U12(2, z') :|: z = 2, z' >= 0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(a__length(mark(z' - 1))) + MARK(1 + (z' - 1)) :|: z = 2, z' - 1 >= 0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(a__U12(mark(z0''), z1'')) + MARK(1 + z0'' + z1'') :|: z = 2, z0'' >= 0, z' = 1 + z0'' + z1'', z1'' >= 0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(a__U11(mark(z0'), z1')) + MARK(1 + z0' + z1') :|: z = 2, z0' >= 0, z1' >= 0, z' = 1 + z0' + z1' A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(3) + MARK(3) :|: z = 2, z' = 3 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(2) + MARK(2) :|: z = 2, z' = 2 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(1) + MARK(1) :|: z = 2, z' = 1 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(0) + MARK(z') :|: z = 2, z' >= 0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(0) + MARK(3) :|: z = 2, z' = 3 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(0) + MARK(0) :|: z = 2, z' = 0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(1 + mark(z' - 1)) + MARK(1 + (z' - 1)) :|: z = 2, z' - 1 >= 0 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(1 + mark(z02) + z11) + MARK(1 + z02 + z11) :|: z = 2, z11 >= 0, z02 >= 0, z' = 1 + z02 + z11 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(1 + 0 + 3) + MARK(3) :|: z = 2, z' = 3 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z - 1) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(a__length(mark(z011)), z1) + MARK(1 + z011) :|: z011 >= 0, z1 >= 0, z = 1 + (1 + z011) + z1 MARK(z) -{ 1 }-> 1 + A__U12(a__U12(mark(z010), z16), z1) + MARK(1 + z010 + z16) :|: z1 >= 0, z = 1 + (1 + z010 + z16) + z1, z16 >= 0, z010 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(a__U11(mark(z09), z15), z1) + MARK(1 + z09 + z15) :|: z15 >= 0, z1 >= 0, z = 1 + (1 + z09 + z15) + z1, z09 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(3, z - 4) + MARK(3) :|: z - 4 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(2, z - 3) + MARK(2) :|: z - 3 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(1, z - 2) + MARK(1) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(0, z1) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__U12(0, z - 4) + MARK(3) :|: z - 4 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(0, z - 1) + MARK(0) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(1 + mark(z013), z1) + MARK(1 + z013) :|: z1 >= 0, z013 >= 0, z = 1 + (1 + z013) + z1 MARK(z) -{ 1 }-> 1 + A__U12(1 + mark(z012) + z17, z1) + MARK(1 + z012 + z17) :|: z1 >= 0, z = 1 + (1 + z012 + z17) + z1, z17 >= 0, z012 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(1 + 0 + 3, z - 4) + MARK(3) :|: z - 4 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(a__length(mark(z06)), z1) + MARK(1 + z06) :|: z1 >= 0, z = 1 + (1 + z06) + z1, z06 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(a__U12(mark(z05), z13), z1) + MARK(1 + z05 + z13) :|: z1 >= 0, z05 >= 0, z = 1 + (1 + z05 + z13) + z1, z13 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(a__U11(mark(z04), z12), z1) + MARK(1 + z04 + z12) :|: z04 >= 0, z = 1 + (1 + z04 + z12) + z1, z1 >= 0, z12 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(3, z - 4) + MARK(3) :|: z - 4 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(2, z - 3) + MARK(2) :|: z - 3 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(1, z - 2) + MARK(1) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(0, z1) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__U11(0, z - 4) + MARK(3) :|: z - 4 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(0, z - 1) + MARK(0) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(1 + mark(z08), z1) + MARK(1 + z08) :|: z08 >= 0, z1 >= 0, z = 1 + (1 + z08) + z1 MARK(z) -{ 1 }-> 1 + A__U11(1 + mark(z07) + z14, z1) + MARK(1 + z07 + z14) :|: z1 >= 0, z07 >= 0, z = 1 + (1 + z07 + z14) + z1, z14 >= 0 MARK(z) -{ 1 }-> 1 + A__U11(1 + 0 + 3, z - 4) + MARK(3) :|: z - 4 >= 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(a__length(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(a__U12(mark(z015), z19)) + MARK(1 + z015 + z19) :|: z015 >= 0, z19 >= 0, z = 1 + (1 + z015 + z19) MARK(z) -{ 1 }-> 1 + A__LENGTH(a__U11(mark(z014), z18)) + MARK(1 + z014 + z18) :|: z = 1 + (1 + z014 + z18), z18 >= 0, z014 >= 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__LENGTH(2) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__LENGTH(1) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__LENGTH(0) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__LENGTH(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(0) + MARK(z - 1) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(1 + mark(z - 2)) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(1 + mark(z017) + z110) + MARK(1 + z017 + z110) :|: z110 >= 0, z = 1 + (1 + z017 + z110), z017 >= 0 MARK(z) -{ 1 }-> 1 + A__LENGTH(1 + 0 + 3) + MARK(3) :|: z = 1 + 3 a__U11(z, z') -{ 0 }-> a__U12(2, z') :|: z = 2, z' >= 0 a__U11(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 a__U11(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0 a__U12(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 a__U12(z, z') -{ 0 }-> 1 + a__length(a__length(mark(z' - 1))) :|: z = 2, z' - 1 >= 0 a__U12(z, z') -{ 0 }-> 1 + a__length(a__U12(mark(z020), z112)) :|: z = 2, z' = 1 + z020 + z112, z112 >= 0, z020 >= 0 a__U12(z, z') -{ 0 }-> 1 + a__length(a__U11(mark(z019), z111)) :|: z = 2, z019 >= 0, z' = 1 + z019 + z111, z111 >= 0 a__U12(z, z') -{ 0 }-> 1 + a__length(3) :|: z = 2, z' = 3 a__U12(z, z') -{ 0 }-> 1 + a__length(2) :|: z = 2, z' = 2 a__U12(z, z') -{ 0 }-> 1 + a__length(1) :|: z = 2, z' = 1 a__U12(z, z') -{ 0 }-> 1 + a__length(0) :|: z = 2, z' = 0 a__U12(z, z') -{ 0 }-> 1 + a__length(0) :|: z = 2, z' >= 0 a__U12(z, z') -{ 0 }-> 1 + a__length(0) :|: z = 2, z' = 3 a__U12(z, z') -{ 0 }-> 1 + a__length(1 + mark(z' - 1)) :|: z = 2, z' - 1 >= 0 a__U12(z, z') -{ 0 }-> 1 + a__length(1 + mark(z022) + z113) :|: z = 2, z022 >= 0, z113 >= 0, z' = 1 + z022 + z113 a__U12(z, z') -{ 0 }-> 1 + a__length(1 + 0 + 3) :|: z = 2, z' = 3 a__U12(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0 a__length(z) -{ 0 }-> a__U11(2, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__length(z) -{ 0 }-> 0 :|: z = 1 a__length(z) -{ 0 }-> 0 :|: z >= 0 a__length(z) -{ 0 }-> 1 + z :|: z >= 0 a__zeros -{ 0 }-> 3 :|: a__zeros -{ 0 }-> 0 :|: a__zeros -{ 0 }-> 1 + 0 + 3 :|: mark(z) -{ 0 }-> a__length(a__length(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__length(a__U12(mark(z035), z121)) :|: z035 >= 0, z = 1 + (1 + z035 + z121), z121 >= 0 mark(z) -{ 0 }-> a__length(a__U11(mark(z034), z120)) :|: z120 >= 0, z = 1 + (1 + z034 + z120), z034 >= 0 mark(z) -{ 0 }-> a__length(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__length(2) :|: z = 1 + 2 mark(z) -{ 0 }-> a__length(1) :|: z = 1 + 1 mark(z) -{ 0 }-> a__length(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__length(0) :|: z - 1 >= 0 mark(z) -{ 0 }-> a__length(0) :|: z = 1 + 3 mark(z) -{ 0 }-> a__length(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__length(1 + mark(z037) + z122) :|: z = 1 + (1 + z037 + z122), z122 >= 0, z037 >= 0 mark(z) -{ 0 }-> a__length(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__U12(a__length(mark(z031)), z1) :|: z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0 mark(z) -{ 0 }-> a__U12(a__U12(mark(z030), z118), z1) :|: z1 >= 0, z118 >= 0, z = 1 + (1 + z030 + z118) + z1, z030 >= 0 mark(z) -{ 0 }-> a__U12(a__U11(mark(z029), z117), z1) :|: z029 >= 0, z117 >= 0, z1 >= 0, z = 1 + (1 + z029 + z117) + z1 mark(z) -{ 0 }-> a__U12(3, z - 4) :|: z - 4 >= 0 mark(z) -{ 0 }-> a__U12(2, z - 3) :|: z - 3 >= 0 mark(z) -{ 0 }-> a__U12(1, z - 2) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__U12(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> a__U12(0, z - 4) :|: z - 4 >= 0 mark(z) -{ 0 }-> a__U12(0, z - 1) :|: z - 1 >= 0 mark(z) -{ 0 }-> a__U12(1 + mark(z033), z1) :|: z1 >= 0, z = 1 + (1 + z033) + z1, z033 >= 0 mark(z) -{ 0 }-> a__U12(1 + mark(z032) + z119, z1) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032 + z119) + z1, z119 >= 0 mark(z) -{ 0 }-> a__U12(1 + 0 + 3, z - 4) :|: z - 4 >= 0 mark(z) -{ 0 }-> a__U11(a__length(mark(z026)), z1) :|: z1 >= 0, z = 1 + (1 + z026) + z1, z026 >= 0 mark(z) -{ 0 }-> a__U11(a__U12(mark(z025), z115), z1) :|: z025 >= 0, z1 >= 0, z115 >= 0, z = 1 + (1 + z025 + z115) + z1 mark(z) -{ 0 }-> a__U11(a__U11(mark(z024), z114), z1) :|: z114 >= 0, z1 >= 0, z024 >= 0, z = 1 + (1 + z024 + z114) + z1 mark(z) -{ 0 }-> a__U11(3, z - 4) :|: z - 4 >= 0 mark(z) -{ 0 }-> a__U11(2, z - 3) :|: z - 3 >= 0 mark(z) -{ 0 }-> a__U11(1, z - 2) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__U11(0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> a__U11(0, z - 4) :|: z - 4 >= 0 mark(z) -{ 0 }-> a__U11(0, z - 1) :|: z - 1 >= 0 mark(z) -{ 0 }-> a__U11(1 + mark(z028), z1) :|: z1 >= 0, z = 1 + (1 + z028) + z1, z028 >= 0 mark(z) -{ 0 }-> a__U11(1 + mark(z027) + z116, z1) :|: z1 >= 0, z027 >= 0, z = 1 + (1 + z027 + z116) + z1, z116 >= 0 mark(z) -{ 0 }-> a__U11(1 + 0 + 3, z - 4) :|: z - 4 >= 0 mark(z) -{ 0 }-> 3 :|: z = 3 mark(z) -{ 0 }-> 2 :|: z = 2 mark(z) -{ 0 }-> 1 :|: z = 1 mark(z) -{ 0 }-> 0 :|: z = 0 mark(z) -{ 0 }-> 0 :|: z >= 0 mark(z) -{ 0 }-> 0 :|: z = 3 mark(z) -{ 0 }-> 1 + mark(z - 1) :|: z - 1 >= 0 mark(z) -{ 0 }-> 1 + mark(z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> 1 + 0 + 3 :|: z = 3 ---------------------------------------- (45) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: a__zeros -> null_a__zeros [0] a__U11(v0, v1) -> null_a__U11 [0] a__U12(v0, v1) -> null_a__U12 [0] a__length(v0) -> null_a__length [0] mark(v0) -> null_mark [0] A__U11(v0, v1) -> null_A__U11 [0] A__U12(v0, v1) -> null_A__U12 [0] A__LENGTH(v0) -> null_A__LENGTH [0] MARK(v0) -> null_MARK [0] And the following fresh constants: null_a__zeros, null_a__U11, null_a__U12, null_a__length, null_mark, null_A__U11, null_A__U12, null_A__LENGTH, null_MARK ---------------------------------------- (46) 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__U11(tt, z0) -> c2(A__U12(tt, z0)) [1] A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) [1] A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) [1] MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) [1] MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) [1] MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) [1] MARK(cons(z0, z1)) -> c13(MARK(z0)) [1] MARK(s(z0)) -> c16(MARK(z0)) [1] a__zeros -> cons(0, zeros) [0] a__zeros -> zeros [0] a__U11(tt, z0) -> a__U12(tt, z0) [0] a__U11(z0, z1) -> U11(z0, z1) [0] a__U12(tt, z0) -> s(a__length(mark(z0))) [0] a__U12(z0, z1) -> U12(z0, z1) [0] a__length(nil) -> 0 [0] a__length(cons(z0, z1)) -> a__U11(tt, z1) [0] a__length(z0) -> length(z0) [0] mark(zeros) -> a__zeros [0] mark(U11(z0, z1)) -> a__U11(mark(z0), z1) [0] mark(U12(z0, z1)) -> a__U12(mark(z0), z1) [0] mark(length(z0)) -> a__length(mark(z0)) [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(0) -> 0 [0] mark(tt) -> tt [0] mark(s(z0)) -> s(mark(z0)) [0] mark(nil) -> nil [0] a__zeros -> null_a__zeros [0] a__U11(v0, v1) -> null_a__U11 [0] a__U12(v0, v1) -> null_a__U12 [0] a__length(v0) -> null_a__length [0] mark(v0) -> null_mark [0] A__U11(v0, v1) -> null_A__U11 [0] A__U12(v0, v1) -> null_A__U12 [0] A__LENGTH(v0) -> null_A__LENGTH [0] MARK(v0) -> null_MARK [0] The TRS has the following type information: A__U11 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> c2:null_A__U11 tt :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c2 :: c4:null_A__U12 -> c2:null_A__U11 A__U12 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> c4:null_A__U12 c4 :: c7:null_A__LENGTH -> c10:c11:c12:c13:c16:null_MARK -> c4:null_A__U12 A__LENGTH :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> c7:null_A__LENGTH mark :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark MARK :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> c10:c11:c12:c13:c16:null_MARK cons :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c7 :: c2:null_A__U11 -> c7:null_A__LENGTH U11 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c10 :: c2:null_A__U11 -> c10:c11:c12:c13:c16:null_MARK -> c10:c11:c12:c13:c16:null_MARK U12 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c11 :: c4:null_A__U12 -> c10:c11:c12:c13:c16:null_MARK -> c10:c11:c12:c13:c16:null_MARK length :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c12 :: c7:null_A__LENGTH -> c10:c11:c12:c13:c16:null_MARK -> c10:c11:c12:c13:c16:null_MARK c13 :: c10:c11:c12:c13:c16:null_MARK -> c10:c11:c12:c13:c16:null_MARK s :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark c16 :: c10:c11:c12:c13:c16:null_MARK -> c10:c11:c12:c13:c16:null_MARK a__zeros :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark 0 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark zeros :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark a__U11 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark a__U12 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark a__length :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark -> tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark nil :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_a__zeros :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_a__U11 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_a__U12 :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_a__length :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_mark :: tt:cons:U11:U12:length:s:0:zeros:nil:null_a__zeros:null_a__U11:null_a__U12:null_a__length:null_mark null_A__U11 :: c2:null_A__U11 null_A__U12 :: c4:null_A__U12 null_A__LENGTH :: c7:null_A__LENGTH null_MARK :: c10:c11:c12:c13:c16:null_MARK Rewrite Strategy: INNERMOST ---------------------------------------- (47) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: tt => 2 0 => 0 zeros => 3 nil => 1 null_a__zeros => 0 null_a__U11 => 0 null_a__U12 => 0 null_a__length => 0 null_mark => 0 null_A__U11 => 0 null_A__U12 => 0 null_A__LENGTH => 0 null_MARK => 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: A__LENGTH(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 A__LENGTH(z) -{ 1 }-> 1 + A__U11(2, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__U11(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 A__U11(z, z') -{ 1 }-> 1 + A__U12(2, z0) :|: z = 2, z0 >= 0, z' = z0 A__U12(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 A__U12(z, z') -{ 1 }-> 1 + A__LENGTH(mark(z0)) + MARK(z0) :|: z = 2, z0 >= 0, z' = z0 MARK(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__U12(mark(z0), z1) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__U11(mark(z0), z1) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__LENGTH(mark(z0)) + MARK(z0) :|: z = 1 + z0, z0 >= 0 a__U11(z, z') -{ 0 }-> a__U12(2, z0) :|: z = 2, z0 >= 0, z' = z0 a__U11(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a__U11(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 a__U12(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a__U12(z, z') -{ 0 }-> 1 + a__length(mark(z0)) :|: z = 2, z0 >= 0, z' = z0 a__U12(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 a__length(z) -{ 0 }-> a__U11(2, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__length(z) -{ 0 }-> 0 :|: z = 1 a__length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__length(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__zeros -{ 0 }-> 3 :|: a__zeros -{ 0 }-> 0 :|: a__zeros -{ 0 }-> 1 + 0 + 3 :|: mark(z) -{ 0 }-> a__zeros :|: z = 3 mark(z) -{ 0 }-> a__length(mark(z0)) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__U12(mark(z0), z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> a__U11(mark(z0), z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> 2 :|: z = 2 mark(z) -{ 0 }-> 1 :|: z = 1 mark(z) -{ 0 }-> 0 :|: z = 0 mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 mark(z) -{ 0 }-> 1 + mark(z0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> 1 + mark(z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (49) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U12(tt, z0) -> c4(A__LENGTH(mark(z0)), MARK(z0)) by A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros), MARK(zeros)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, 0) -> c4(A__LENGTH(0), MARK(0)) A__U12(tt, tt) -> c4(A__LENGTH(tt), MARK(tt)) A__U12(tt, s(z0)) -> c4(A__LENGTH(s(mark(z0))), MARK(s(z0))) A__U12(tt, nil) -> c4(A__LENGTH(nil), MARK(nil)) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros), MARK(zeros)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, 0) -> c4(A__LENGTH(0), MARK(0)) A__U12(tt, tt) -> c4(A__LENGTH(tt), MARK(tt)) A__U12(tt, s(z0)) -> c4(A__LENGTH(s(mark(z0))), MARK(s(z0))) A__U12(tt, nil) -> c4(A__LENGTH(nil), MARK(nil)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros), MARK(zeros)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, 0) -> c4(A__LENGTH(0), MARK(0)) A__U12(tt, tt) -> c4(A__LENGTH(tt), MARK(tt)) A__U12(tt, s(z0)) -> c4(A__LENGTH(s(mark(z0))), MARK(s(z0))) A__U12(tt, nil) -> c4(A__LENGTH(nil), MARK(nil)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c10_2, c11_2, c12_2, c13_1, c16_1, c4_2 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: A__U12(tt, nil) -> c4(A__LENGTH(nil), MARK(nil)) A__U12(tt, 0) -> c4(A__LENGTH(0), MARK(0)) A__U12(tt, tt) -> c4(A__LENGTH(tt), MARK(tt)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros), MARK(zeros)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(A__LENGTH(s(mark(z0))), MARK(s(z0))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros), MARK(zeros)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(A__LENGTH(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c10_2, c11_2, c12_2, c13_1, c16_1, c4_2 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c10_2, c11_2, c12_2, c13_1, c16_1, c4_2, c4_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(z0, z1)) -> c10(A__U11(mark(z0), z1), MARK(z0)) by MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1), MARK(zeros)) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(A__U11(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(U11(0, x1)) -> c10(A__U11(0, x1), MARK(0)) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1), MARK(tt)) MARK(U11(s(z0), x1)) -> c10(A__U11(s(mark(z0)), x1), MARK(s(z0))) MARK(U11(nil, x1)) -> c10(A__U11(nil, x1), MARK(nil)) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1), MARK(zeros)) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(A__U11(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(U11(0, x1)) -> c10(A__U11(0, x1), MARK(0)) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1), MARK(tt)) MARK(U11(s(z0), x1)) -> c10(A__U11(s(mark(z0)), x1), MARK(s(z0))) MARK(U11(nil, x1)) -> c10(A__U11(nil, x1), MARK(nil)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1), MARK(zeros)) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(A__U11(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(U11(0, x1)) -> c10(A__U11(0, x1), MARK(0)) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1), MARK(tt)) MARK(U11(s(z0), x1)) -> c10(A__U11(s(mark(z0)), x1), MARK(s(z0))) MARK(U11(nil, x1)) -> c10(A__U11(nil, x1), MARK(nil)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c11_2, c12_2, c13_1, c16_1, c4_2, c4_1, c10_2 ---------------------------------------- (57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(U11(0, x1)) -> c10(A__U11(0, x1), MARK(0)) MARK(U11(nil, x1)) -> c10(A__U11(nil, x1), MARK(nil)) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1), MARK(zeros)) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(A__U11(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1), MARK(tt)) MARK(U11(s(z0), x1)) -> c10(A__U11(s(mark(z0)), x1), MARK(s(z0))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1), MARK(zeros)) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(A__U11(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1), MARK(tt)) MARK(U11(s(z0), x1)) -> c10(A__U11(s(mark(z0)), x1), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c11_2, c12_2, c13_1, c16_1, c4_2, c4_1, c10_2 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c11_2, c12_2, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U12(z0, z1)) -> c11(A__U12(mark(z0), z1), MARK(z0)) by MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1), MARK(zeros)) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(A__U12(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(U12(0, x1)) -> c11(A__U12(0, x1), MARK(0)) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1), MARK(tt)) MARK(U12(s(z0), x1)) -> c11(A__U12(s(mark(z0)), x1), MARK(s(z0))) MARK(U12(nil, x1)) -> c11(A__U12(nil, x1), MARK(nil)) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1), MARK(zeros)) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(A__U12(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(U12(0, x1)) -> c11(A__U12(0, x1), MARK(0)) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1), MARK(tt)) MARK(U12(s(z0), x1)) -> c11(A__U12(s(mark(z0)), x1), MARK(s(z0))) MARK(U12(nil, x1)) -> c11(A__U12(nil, x1), MARK(nil)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1), MARK(zeros)) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(A__U12(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(U12(0, x1)) -> c11(A__U12(0, x1), MARK(0)) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1), MARK(tt)) MARK(U12(s(z0), x1)) -> c11(A__U12(s(mark(z0)), x1), MARK(s(z0))) MARK(U12(nil, x1)) -> c11(A__U12(nil, x1), MARK(nil)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c12_2, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2 ---------------------------------------- (63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(U12(0, x1)) -> c11(A__U12(0, x1), MARK(0)) MARK(U12(nil, x1)) -> c11(A__U12(nil, x1), MARK(nil)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1), MARK(zeros)) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(A__U12(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1), MARK(tt)) MARK(U12(s(z0), x1)) -> c11(A__U12(s(mark(z0)), x1), MARK(s(z0))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1), MARK(zeros)) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(A__U12(cons(mark(z0), z1), x1), MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1), MARK(tt)) MARK(U12(s(z0), x1)) -> c11(A__U12(s(mark(z0)), x1), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c12_2, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c12_2, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(z0)) -> c12(A__LENGTH(mark(z0)), MARK(z0)) by MARK(length(zeros)) -> c12(A__LENGTH(a__zeros), MARK(zeros)) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(0)) -> c12(A__LENGTH(0), MARK(0)) MARK(length(tt)) -> c12(A__LENGTH(tt), MARK(tt)) MARK(length(s(z0))) -> c12(A__LENGTH(s(mark(z0))), MARK(s(z0))) MARK(length(nil)) -> c12(A__LENGTH(nil), MARK(nil)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros), MARK(zeros)) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(0)) -> c12(A__LENGTH(0), MARK(0)) MARK(length(tt)) -> c12(A__LENGTH(tt), MARK(tt)) MARK(length(s(z0))) -> c12(A__LENGTH(s(mark(z0))), MARK(s(z0))) MARK(length(nil)) -> c12(A__LENGTH(nil), MARK(nil)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros), MARK(zeros)) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(0)) -> c12(A__LENGTH(0), MARK(0)) MARK(length(tt)) -> c12(A__LENGTH(tt), MARK(tt)) MARK(length(s(z0))) -> c12(A__LENGTH(s(mark(z0))), MARK(s(z0))) MARK(length(nil)) -> c12(A__LENGTH(nil), MARK(nil)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1, c12_2 ---------------------------------------- (69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(length(tt)) -> c12(A__LENGTH(tt), MARK(tt)) MARK(length(nil)) -> c12(A__LENGTH(nil), MARK(nil)) MARK(length(0)) -> c12(A__LENGTH(0), MARK(0)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros), MARK(zeros)) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(A__LENGTH(s(mark(z0))), MARK(s(z0))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros), MARK(zeros)) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(A__LENGTH(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1, c12_2 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U12(tt, U11(z0, z1)) -> c4(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) by A__U12(tt, U11(x0, z1)) -> c4(A__LENGTH(U11(mark(x0), z1)), MARK(U11(x0, z1))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1)), MARK(U11(0, x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1)), MARK(U11(nil, x1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(x0, z1)) -> c4(A__LENGTH(U11(mark(x0), z1)), MARK(U11(x0, z1))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1)), MARK(U11(0, x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1)), MARK(U11(nil, x1))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(x0, z1)) -> c4(A__LENGTH(U11(mark(x0), z1)), MARK(U11(x0, z1))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1)), MARK(U11(0, x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1)), MARK(U11(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U12(tt, U12(z0, z1)) -> c4(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) by A__U12(tt, U12(x0, z1)) -> c4(A__LENGTH(U12(mark(x0), z1)), MARK(U12(x0, z1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1)), MARK(U12(0, x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1)), MARK(U12(nil, x1))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(x0, z1)) -> c4(A__LENGTH(U12(mark(x0), z1)), MARK(U12(x0, z1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1)), MARK(U12(0, x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1)), MARK(U12(nil, x1))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(x0, z1)) -> c4(A__LENGTH(U12(mark(x0), z1)), MARK(U12(x0, z1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1)), MARK(U12(0, x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1)), MARK(U12(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U12(tt, length(z0)) -> c4(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) by A__U12(tt, length(x0)) -> c4(A__LENGTH(length(mark(x0))), MARK(length(x0))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0)), MARK(length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt)), MARK(length(tt))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil)), MARK(length(nil))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(x0)) -> c4(A__LENGTH(length(mark(x0))), MARK(length(x0))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0)), MARK(length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt)), MARK(length(tt))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil)), MARK(length(nil))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(x0)) -> c4(A__LENGTH(length(mark(x0))), MARK(length(x0))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0)), MARK(length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt)), MARK(length(tt))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil)), MARK(length(nil))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U12(tt, zeros) -> c4(A__LENGTH(a__zeros)) by A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) A__U12(tt, zeros) -> c4(A__LENGTH(zeros)) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) A__U12(tt, zeros) -> c4(A__LENGTH(zeros)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) A__U12(tt, zeros) -> c4(A__LENGTH(zeros)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U12(tt, zeros) -> c4(A__LENGTH(zeros)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(U11(z0, z1), x1)) -> c10(A__U11(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) by MARK(U11(U11(x0, z1), x2)) -> c10(A__U11(U11(mark(x0), z1), x2), MARK(U11(x0, z1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2), MARK(U11(0, x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2), MARK(U11(nil, x1))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(x0, z1), x2)) -> c10(A__U11(U11(mark(x0), z1), x2), MARK(U11(x0, z1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2), MARK(U11(0, x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2), MARK(U11(nil, x1))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(x0, z1), x2)) -> c10(A__U11(U11(mark(x0), z1), x2), MARK(U11(x0, z1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2), MARK(U11(0, x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2), MARK(U11(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (91) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(U12(z0, z1), x1)) -> c10(A__U11(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) by MARK(U11(U12(x0, z1), x2)) -> c10(A__U11(U12(mark(x0), z1), x2), MARK(U12(x0, z1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2), MARK(U12(0, x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2), MARK(U12(nil, x1))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(x0, z1), x2)) -> c10(A__U11(U12(mark(x0), z1), x2), MARK(U12(x0, z1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2), MARK(U12(0, x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2), MARK(U12(nil, x1))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(x0, z1), x2)) -> c10(A__U11(U12(mark(x0), z1), x2), MARK(U12(x0, z1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2), MARK(U12(0, x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2), MARK(U12(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (95) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(length(z0), x1)) -> c10(A__U11(a__length(mark(z0)), x1), MARK(length(z0))) by MARK(U11(length(x0), x1)) -> c10(A__U11(length(mark(x0)), x1), MARK(length(x0))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1), MARK(length(0))) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1), MARK(length(tt))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1), MARK(length(nil))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(x0), x1)) -> c10(A__U11(length(mark(x0)), x1), MARK(length(x0))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1), MARK(length(0))) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1), MARK(length(tt))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1), MARK(length(nil))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(x0), x1)) -> c10(A__U11(length(mark(x0)), x1), MARK(length(x0))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1), MARK(length(0))) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1), MARK(length(tt))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1), MARK(length(nil))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_2, c11_1, c12_2, c12_1, c10_2 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_2, c11_1, c12_2, c12_1, c10_2 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(zeros, x1)) -> c10(A__U11(a__zeros, x1)) by MARK(U11(zeros, x0)) -> c10(A__U11(cons(0, zeros), x0)) MARK(U11(zeros, x0)) -> c10(A__U11(zeros, x0)) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) MARK(U11(zeros, x0)) -> c10(A__U11(cons(0, zeros), x0)) MARK(U11(zeros, x0)) -> c10(A__U11(zeros, x0)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) MARK(U11(zeros, x0)) -> c10(A__U11(cons(0, zeros), x0)) MARK(U11(zeros, x0)) -> c10(A__U11(zeros, x0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_2, c11_1, c12_2, c12_1, c10_2 ---------------------------------------- (103) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(U11(zeros, x0)) -> c10(A__U11(zeros, x0)) MARK(U11(zeros, x0)) -> c10(A__U11(cons(0, zeros), x0)) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_2, c11_1, c12_2, c12_1, c10_2 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_2, c11_1, c12_2, c12_1, c10_2 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U12(U11(z0, z1), x1)) -> c11(A__U12(a__U11(mark(z0), z1), x1), MARK(U11(z0, z1))) by MARK(U12(U11(x0, z1), x2)) -> c11(A__U12(U11(mark(x0), z1), x2), MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2), MARK(U11(0, x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2), MARK(U11(nil, x1))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(x0, z1), x2)) -> c11(A__U12(U11(mark(x0), z1), x2), MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2), MARK(U11(0, x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2), MARK(U11(nil, x1))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(x0, z1), x2)) -> c11(A__U12(U11(mark(x0), z1), x2), MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2), MARK(U11(zeros, x1))) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2), MARK(U11(0, x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2), MARK(U11(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_2, c11_1, c12_2, c12_1, c10_2 ---------------------------------------- (109) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_2, c11_1, c12_2, c12_1, c10_2 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U12(U12(z0, z1), x1)) -> c11(A__U12(a__U12(mark(z0), z1), x1), MARK(U12(z0, z1))) by MARK(U12(U12(x0, z1), x2)) -> c11(A__U12(U12(mark(x0), z1), x2), MARK(U12(x0, z1))) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2), MARK(U12(0, x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2), MARK(U12(nil, x1))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(x0, z1), x2)) -> c11(A__U12(U12(mark(x0), z1), x2), MARK(U12(x0, z1))) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2), MARK(U12(0, x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2), MARK(U12(nil, x1))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(x0, z1), x2)) -> c11(A__U12(U12(mark(x0), z1), x2), MARK(U12(x0, z1))) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2), MARK(U12(0, x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2), MARK(U12(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_2, c11_1, c12_2, c12_1, c10_2 ---------------------------------------- (113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_2, c11_1, c12_2, c12_1, c10_2 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U12(length(z0), x1)) -> c11(A__U12(a__length(mark(z0)), x1), MARK(length(z0))) by MARK(U12(length(x0), x1)) -> c11(A__U12(length(mark(x0)), x1), MARK(length(x0))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1), MARK(length(0))) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1), MARK(length(tt))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1), MARK(length(nil))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(x0), x1)) -> c11(A__U12(length(mark(x0)), x1), MARK(length(x0))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1), MARK(length(0))) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1), MARK(length(tt))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1), MARK(length(nil))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(x0), x1)) -> c11(A__U12(length(mark(x0)), x1), MARK(length(x0))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1), MARK(length(0))) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1), MARK(length(tt))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1), MARK(length(nil))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U12(zeros, x1)) -> c11(A__U12(a__zeros, x1)) by MARK(U12(zeros, x0)) -> c11(A__U12(cons(0, zeros), x0)) MARK(U12(zeros, x0)) -> c11(A__U12(zeros, x0)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) MARK(U12(zeros, x0)) -> c11(A__U12(cons(0, zeros), x0)) MARK(U12(zeros, x0)) -> c11(A__U12(zeros, x0)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) MARK(U12(zeros, x0)) -> c11(A__U12(cons(0, zeros), x0)) MARK(U12(zeros, x0)) -> c11(A__U12(zeros, x0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(U12(zeros, x0)) -> c11(A__U12(zeros, x0)) MARK(U12(zeros, x0)) -> c11(A__U12(cons(0, zeros), x0)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2), MARK(U12(zeros, x1))) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (123) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(U11(z0, z1))) -> c12(A__LENGTH(a__U11(mark(z0), z1)), MARK(U11(z0, z1))) by MARK(length(U11(x0, z1))) -> c12(A__LENGTH(U11(mark(x0), z1)), MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1)), MARK(U11(0, x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1)), MARK(U11(nil, x1))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> 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MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(x0, z1))) -> c12(A__LENGTH(U11(mark(x0), z1)), MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1)), MARK(U11(0, x1))) MARK(length(U11(tt, 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A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(x0, z1))) -> c12(A__LENGTH(U11(mark(x0), z1)), MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1)), MARK(U11(zeros, x1))) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1)), MARK(U11(0, x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1)), MARK(U11(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (127) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (129) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(U12(z0, z1))) -> c12(A__LENGTH(a__U12(mark(z0), z1)), MARK(U12(z0, z1))) by MARK(length(U12(x0, z1))) -> c12(A__LENGTH(U12(mark(x0), z1)), MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1)), MARK(U12(0, x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1)), MARK(U12(nil, x1))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(x0, z1))) -> c12(A__LENGTH(U12(mark(x0), z1)), MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1)), MARK(U12(0, x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1)), MARK(U12(nil, x1))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(x0, z1))) -> c12(A__LENGTH(U12(mark(x0), z1)), MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1)), MARK(U12(zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1)), MARK(U12(0, x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1)), MARK(U12(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (131) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, 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c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(length(z0))) -> c12(A__LENGTH(a__length(mark(z0))), MARK(length(z0))) by MARK(length(length(x0))) -> c12(A__LENGTH(length(mark(x0))), MARK(length(x0))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0)), MARK(length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt)), MARK(length(tt))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil)), MARK(length(nil))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> 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c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(x0))) -> c12(A__LENGTH(length(mark(x0))), MARK(length(x0))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0)), MARK(length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt)), MARK(length(tt))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil)), MARK(length(nil))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (135) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> 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c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), 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z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(zeros)) -> c12(A__LENGTH(a__zeros)) by MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(length(zeros)) -> c12(A__LENGTH(zeros)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> 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MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(length(zeros)) -> c12(A__LENGTH(zeros)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (139) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(length(zeros)) -> c12(A__LENGTH(zeros)) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (141) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U12(tt, U11(0, x1)) -> c4(A__LENGTH(a__U11(0, x1))) by A__U12(tt, U11(0, z1)) -> c4(A__LENGTH(U11(0, z1))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) A__U12(tt, U11(0, z1)) -> c4(A__LENGTH(U11(0, z1))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) A__U12(tt, U11(0, z1)) -> c4(A__LENGTH(U11(0, z1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (143) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U12(tt, U11(0, z1)) -> c4(A__LENGTH(U11(0, z1))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> 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MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (145) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U12(tt, U11(nil, x1)) -> c4(A__LENGTH(a__U11(nil, x1))) by A__U12(tt, U11(nil, z1)) -> c4(A__LENGTH(U11(nil, z1))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> 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MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> 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U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) A__U12(tt, U11(nil, z1)) -> c4(A__LENGTH(U11(nil, z1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U12(tt, U11(nil, z1)) -> c4(A__LENGTH(U11(nil, z1))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) 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c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> 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MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (149) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U12(tt, U12(0, x1)) -> c4(A__LENGTH(a__U12(0, x1))) by A__U12(tt, U12(0, z1)) -> c4(A__LENGTH(U12(0, z1))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) A__U12(tt, U12(0, z1)) -> c4(A__LENGTH(U12(0, z1))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) A__U12(tt, U12(0, z1)) -> c4(A__LENGTH(U12(0, z1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (151) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U12(tt, U12(0, z1)) -> c4(A__LENGTH(U12(0, z1))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (153) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U12(tt, U12(nil, x1)) -> c4(A__LENGTH(a__U12(nil, x1))) by A__U12(tt, U12(nil, z1)) -> c4(A__LENGTH(U12(nil, z1))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), 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c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, 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MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) A__U12(tt, U12(nil, z1)) -> c4(A__LENGTH(U12(nil, z1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (155) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U12(tt, U12(nil, z1)) -> c4(A__LENGTH(U12(nil, z1))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) 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MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (157) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U12(tt, length(0)) -> c4(A__LENGTH(a__length(0))) by A__U12(tt, length(0)) -> c4(A__LENGTH(length(0))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), 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c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) A__U12(tt, length(0)) -> c4(A__LENGTH(length(0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (159) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U12(tt, length(0)) -> c4(A__LENGTH(length(0))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (161) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U12(tt, length(tt)) -> c4(A__LENGTH(a__length(tt))) by A__U12(tt, length(tt)) -> c4(A__LENGTH(length(tt))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) A__U12(tt, length(tt)) -> c4(A__LENGTH(length(tt))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) A__U12(tt, length(tt)) -> c4(A__LENGTH(length(tt))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (163) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U12(tt, length(tt)) -> c4(A__LENGTH(length(tt))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) 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MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (165) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__U12(tt, length(nil)) -> c4(A__LENGTH(a__length(nil))) by A__U12(tt, length(nil)) -> c4(A__LENGTH(0)) A__U12(tt, length(nil)) -> c4(A__LENGTH(length(nil))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) A__U12(tt, length(nil)) -> c4(A__LENGTH(0)) A__U12(tt, length(nil)) -> c4(A__LENGTH(length(nil))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) A__U12(tt, length(nil)) -> c4(A__LENGTH(0)) A__U12(tt, length(nil)) -> c4(A__LENGTH(length(nil))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (167) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: A__U12(tt, length(nil)) -> c4(A__LENGTH(length(nil))) A__U12(tt, length(nil)) -> c4(A__LENGTH(0)) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (169) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(U11(0, x1), x2)) -> c10(A__U11(a__U11(0, x1), x2)) by MARK(U11(U11(0, z1), x1)) -> c10(A__U11(U11(0, z1), x1)) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U11(U11(0, z1), x1)) -> c10(A__U11(U11(0, z1), x1)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U11(U11(0, z1), x1)) -> c10(A__U11(U11(0, z1), x1)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (171) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U11(U11(0, z1), x1)) -> c10(A__U11(U11(0, z1), x1)) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), 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c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S 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x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (173) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(U11(nil, x1), x2)) -> c10(A__U11(a__U11(nil, x1), x2)) by MARK(U11(U11(nil, z1), x1)) -> c10(A__U11(U11(nil, z1), x1)) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), 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x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U11(U11(nil, z1), x1)) -> c10(A__U11(U11(nil, z1), x1)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (175) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U11(U11(nil, z1), x1)) -> c10(A__U11(U11(nil, z1), x1)) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (177) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(U12(0, x1), x2)) -> c10(A__U11(a__U12(0, x1), x2)) by MARK(U11(U12(0, z1), x1)) -> c10(A__U11(U12(0, z1), x1)) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U11(U12(0, z1), x1)) -> c10(A__U11(U12(0, z1), x1)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U11(U12(0, z1), x1)) -> c10(A__U11(U12(0, z1), x1)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (179) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U11(U12(0, z1), x1)) -> c10(A__U11(U12(0, z1), x1)) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) 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MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (181) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(U12(nil, x1), x2)) -> c10(A__U11(a__U12(nil, x1), x2)) by MARK(U11(U12(nil, z1), x1)) -> c10(A__U11(U12(nil, z1), x1)) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U11(U12(nil, z1), x1)) -> c10(A__U11(U12(nil, z1), x1)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U11(U12(nil, z1), x1)) -> c10(A__U11(U12(nil, z1), x1)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (183) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U11(U12(nil, z1), x1)) -> c10(A__U11(U12(nil, z1), x1)) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (185) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(length(0), x1)) -> c10(A__U11(a__length(0), x1)) by MARK(U11(length(0), x0)) -> c10(A__U11(length(0), x0)) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U11(length(0), x0)) -> c10(A__U11(length(0), x0)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U11(length(0), x0)) -> c10(A__U11(length(0), x0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (187) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U11(length(0), x0)) -> c10(A__U11(length(0), x0)) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) 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c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> 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x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (189) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(length(tt), x1)) -> c10(A__U11(a__length(tt), x1)) by MARK(U11(length(tt), x0)) -> c10(A__U11(length(tt), x0)) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U11(length(tt), x0)) -> c10(A__U11(length(tt), x0)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U11(length(tt), x0)) -> c10(A__U11(length(tt), x0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (191) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U11(length(tt), x0)) -> c10(A__U11(length(tt), x0)) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (193) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U11(length(nil), x1)) -> c10(A__U11(a__length(nil), x1)) by MARK(U11(length(nil), x0)) -> c10(A__U11(0, x0)) MARK(U11(length(nil), x0)) -> c10(A__U11(length(nil), x0)) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) 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c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> 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MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U11(length(nil), x0)) -> c10(A__U11(0, x0)) MARK(U11(length(nil), x0)) -> c10(A__U11(length(nil), x0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (195) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(U11(length(nil), x0)) -> c10(A__U11(length(nil), x0)) MARK(U11(length(nil), x0)) -> c10(A__U11(0, x0)) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> 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c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), 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c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (197) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U12(U11(0, x1), x2)) -> c11(A__U12(a__U11(0, x1), x2)) by MARK(U12(U11(0, z1), x1)) -> c11(A__U12(U11(0, z1), x1)) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U12(U11(0, z1), x1)) -> c11(A__U12(U11(0, z1), x1)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U12(U11(0, z1), x1)) -> c11(A__U12(U11(0, z1), x1)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (199) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U12(U11(0, z1), x1)) -> c11(A__U12(U11(0, z1), x1)) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, 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MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), 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c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (201) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U12(U11(nil, x1), x2)) -> c11(A__U12(a__U11(nil, x1), x2)) by MARK(U12(U11(nil, z1), x1)) -> c11(A__U12(U11(nil, z1), x1)) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), 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x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U12(U11(nil, z1), x1)) -> c11(A__U12(U11(nil, z1), x1)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (203) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U12(U11(nil, z1), x1)) -> c11(A__U12(U11(nil, z1), x1)) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (205) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U12(U12(0, x1), x2)) -> c11(A__U12(a__U12(0, x1), x2)) by MARK(U12(U12(0, z1), x1)) -> c11(A__U12(U12(0, z1), x1)) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> 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c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U12(U12(0, z1), x1)) -> c11(A__U12(U12(0, z1), x1)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (207) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U12(U12(0, z1), x1)) -> c11(A__U12(U12(0, z1), x1)) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, 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MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (209) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U12(U12(nil, x1), x2)) -> c11(A__U12(a__U12(nil, x1), x2)) by MARK(U12(U12(nil, z1), x1)) -> c11(A__U12(U12(nil, z1), x1)) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U12(U12(nil, z1), x1)) -> c11(A__U12(U12(nil, z1), x1)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U12(U12(nil, z1), x1)) -> c11(A__U12(U12(nil, z1), x1)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (211) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U12(U12(nil, z1), x1)) -> c11(A__U12(U12(nil, z1), x1)) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) 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x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (213) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U12(length(0), x1)) -> c11(A__U12(a__length(0), x1)) by MARK(U12(length(0), x0)) -> c11(A__U12(length(0), x0)) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) 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MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U12(length(0), x0)) -> c11(A__U12(length(0), x0)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), 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c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U12(length(0), x0)) -> c11(A__U12(length(0), x0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (215) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U12(length(0), x0)) -> c11(A__U12(length(0), x0)) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (217) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U12(length(tt), x1)) -> c11(A__U12(a__length(tt), x1)) by MARK(U12(length(tt), x0)) -> c11(A__U12(length(tt), x0)) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), 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c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U12(length(tt), x0)) -> c11(A__U12(length(tt), x0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (219) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U12(length(tt), x0)) -> c11(A__U12(length(tt), x0)) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (221) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(U12(length(nil), x1)) -> c11(A__U12(a__length(nil), x1)) by MARK(U12(length(nil), x0)) -> c11(A__U12(0, x0)) MARK(U12(length(nil), x0)) -> c11(A__U12(length(nil), x0)) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U12(length(nil), x0)) -> c11(A__U12(0, x0)) MARK(U12(length(nil), x0)) -> c11(A__U12(length(nil), x0)) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(U12(length(nil), x0)) -> c11(A__U12(0, x0)) MARK(U12(length(nil), x0)) -> c11(A__U12(length(nil), x0)) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (223) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(U12(length(nil), x0)) -> c11(A__U12(length(nil), x0)) MARK(U12(length(nil), x0)) -> c11(A__U12(0, x0)) ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) 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c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (225) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(U11(0, x1))) -> c12(A__LENGTH(a__U11(0, x1))) by MARK(length(U11(0, z1))) -> c12(A__LENGTH(U11(0, z1))) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(length(U11(0, z1))) -> c12(A__LENGTH(U11(0, z1))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(length(U11(0, z1))) -> c12(A__LENGTH(U11(0, z1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (227) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(length(U11(0, z1))) -> c12(A__LENGTH(U11(0, z1))) ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (229) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(U11(nil, x1))) -> c12(A__LENGTH(a__U11(nil, x1))) by MARK(length(U11(nil, z1))) -> c12(A__LENGTH(U11(nil, z1))) ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> 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x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(length(U11(nil, z1))) -> c12(A__LENGTH(U11(nil, z1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (231) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(length(U11(nil, z1))) -> c12(A__LENGTH(U11(nil, z1))) ---------------------------------------- (232) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (233) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(U12(0, x1))) -> c12(A__LENGTH(a__U12(0, x1))) by MARK(length(U12(0, z1))) -> c12(A__LENGTH(U12(0, z1))) ---------------------------------------- (234) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), 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z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(length(U12(0, z1))) -> c12(A__LENGTH(U12(0, z1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (235) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(length(U12(0, z1))) -> c12(A__LENGTH(U12(0, z1))) ---------------------------------------- (236) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (237) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(U12(nil, x1))) -> c12(A__LENGTH(a__U12(nil, x1))) by MARK(length(U12(nil, z1))) -> c12(A__LENGTH(U12(nil, z1))) ---------------------------------------- (238) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(length(U12(nil, z1))) -> c12(A__LENGTH(U12(nil, z1))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(length(U12(nil, z1))) -> c12(A__LENGTH(U12(nil, z1))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (239) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(length(U12(nil, z1))) -> c12(A__LENGTH(U12(nil, z1))) ---------------------------------------- (240) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), 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MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (241) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(length(length(0))) -> c12(A__LENGTH(a__length(0))) by MARK(length(length(0))) -> c12(A__LENGTH(length(0))) ---------------------------------------- (242) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(length(length(0))) -> c12(A__LENGTH(length(0))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) MARK(length(length(0))) -> c12(A__LENGTH(length(0))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (243) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(length(length(0))) -> c12(A__LENGTH(length(0))) ---------------------------------------- (244) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> a__U12(tt, z0) a__U11(z0, z1) -> U11(z0, z1) a__U12(tt, z0) -> s(a__length(mark(z0))) a__U12(z0, z1) -> U12(z0, z1) a__length(nil) -> 0 a__length(cons(z0, z1)) -> a__U11(tt, z1) a__length(z0) -> length(z0) mark(zeros) -> a__zeros mark(U11(z0, z1)) -> a__U11(mark(z0), z1) mark(U12(z0, z1)) -> a__U12(mark(z0), z1) mark(length(z0)) -> a__length(mark(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__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) S tuples: A__U11(tt, z0) -> c2(A__U12(tt, z0)) A__LENGTH(cons(z0, z1)) -> c7(A__U11(tt, z1)) MARK(cons(z0, z1)) -> c13(MARK(z0)) MARK(s(z0)) -> c16(MARK(z0)) A__U12(tt, cons(z0, z1)) -> c4(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) A__U12(tt, s(z0)) -> c4(MARK(s(z0))) MARK(U11(cons(z0, z1), x1)) -> c10(MARK(cons(z0, z1))) MARK(U11(tt, x1)) -> c10(A__U11(tt, x1)) MARK(U11(s(z0), x1)) -> c10(MARK(s(z0))) MARK(U12(cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(U12(tt, x1)) -> c11(A__U12(tt, x1)) MARK(U12(s(z0), x1)) -> c11(MARK(s(z0))) MARK(length(cons(z0, z1))) -> c12(A__LENGTH(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(length(s(z0))) -> c12(MARK(s(z0))) A__U12(tt, U11(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) A__U12(tt, U11(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) A__U12(tt, U11(length(z0), x1)) -> c4(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) A__U12(tt, U11(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) A__U12(tt, U11(tt, x1)) -> c4(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) A__U12(tt, U11(s(z0), x1)) -> c4(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) A__U12(tt, U11(x0, z1)) -> c4(MARK(U11(x0, z1))) A__U12(tt, U12(U11(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) A__U12(tt, U12(U12(z0, z1), x1)) -> c4(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) A__U12(tt, U12(length(z0), x1)) -> c4(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) A__U12(tt, U12(cons(z0, z1), x1)) -> c4(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) A__U12(tt, U12(tt, x1)) -> c4(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) A__U12(tt, U12(s(z0), x1)) -> c4(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) A__U12(tt, U12(x0, z1)) -> c4(MARK(U12(x0, z1))) A__U12(tt, length(zeros)) -> c4(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) A__U12(tt, length(U11(z0, z1))) -> c4(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) A__U12(tt, length(U12(z0, z1))) -> c4(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) A__U12(tt, length(length(z0))) -> c4(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) A__U12(tt, length(cons(z0, z1))) -> c4(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) A__U12(tt, length(s(z0))) -> c4(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) A__U12(tt, length(x0)) -> c4(MARK(length(x0))) A__U12(tt, zeros) -> c4(A__LENGTH(cons(0, zeros))) MARK(U11(U11(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U11(U11(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U11(U11(length(z0), x1), x2)) -> c10(A__U11(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U11(U11(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U11(U11(tt, x1), x2)) -> c10(A__U11(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U11(U11(s(z0), x1), x2)) -> c10(A__U11(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U11(U11(x0, z1), x2)) -> c10(MARK(U11(x0, z1))) MARK(U11(U12(U11(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U11(U12(U12(z0, z1), x1), x2)) -> c10(A__U11(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U11(U12(length(z0), x1), x2)) -> c10(A__U11(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U11(U12(cons(z0, z1), x1), x2)) -> c10(A__U11(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U11(U12(tt, x1), x2)) -> c10(A__U11(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U11(U12(s(z0), x1), x2)) -> c10(A__U11(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U11(U12(x0, z1), x2)) -> c10(MARK(U12(x0, z1))) MARK(U11(length(zeros), x1)) -> c10(A__U11(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U11(length(U11(z0, z1)), x1)) -> c10(A__U11(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U11(length(U12(z0, z1)), x1)) -> c10(A__U11(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U11(length(length(z0)), x1)) -> c10(A__U11(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U11(length(cons(z0, z1)), x1)) -> c10(A__U11(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U11(length(s(z0)), x1)) -> c10(A__U11(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U11(length(x0), x1)) -> c10(MARK(length(x0))) A__U12(tt, U11(zeros, x1)) -> c4(A__LENGTH(a__U11(a__zeros, x1))) MARK(U11(U11(zeros, x1), x2)) -> c10(A__U11(a__U11(a__zeros, x1), x2)) MARK(U12(U11(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U11(mark(z0), z1), x1), x2), MARK(U11(U11(z0, z1), x1))) MARK(U12(U11(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U11(a__U12(mark(z0), z1), x1), x2), MARK(U11(U12(z0, z1), x1))) MARK(U12(U11(length(z0), x1), x2)) -> c11(A__U12(a__U11(a__length(mark(z0)), x1), x2), MARK(U11(length(z0), x1))) MARK(U12(U11(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U11(cons(mark(z0), z1), x1), x2), MARK(U11(cons(z0, z1), x1))) MARK(U12(U11(tt, x1), x2)) -> c11(A__U12(a__U11(tt, x1), x2), MARK(U11(tt, x1))) MARK(U12(U11(s(z0), x1), x2)) -> c11(A__U12(a__U11(s(mark(z0)), x1), x2), MARK(U11(s(z0), x1))) MARK(U12(U11(x0, z1), x2)) -> c11(MARK(U11(x0, z1))) MARK(U12(U11(zeros, x1), x2)) -> c11(A__U12(a__U11(a__zeros, x1), x2)) MARK(U12(U12(U11(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U11(mark(z0), z1), x1), x2), MARK(U12(U11(z0, z1), x1))) MARK(U12(U12(U12(z0, z1), x1), x2)) -> c11(A__U12(a__U12(a__U12(mark(z0), z1), x1), x2), MARK(U12(U12(z0, z1), x1))) MARK(U12(U12(length(z0), x1), x2)) -> c11(A__U12(a__U12(a__length(mark(z0)), x1), x2), MARK(U12(length(z0), x1))) MARK(U12(U12(cons(z0, z1), x1), x2)) -> c11(A__U12(a__U12(cons(mark(z0), z1), x1), x2), MARK(U12(cons(z0, z1), x1))) MARK(U12(U12(tt, x1), x2)) -> c11(A__U12(a__U12(tt, x1), x2), MARK(U12(tt, x1))) MARK(U12(U12(s(z0), x1), x2)) -> c11(A__U12(a__U12(s(mark(z0)), x1), x2), MARK(U12(s(z0), x1))) MARK(U12(U12(x0, z1), x2)) -> c11(MARK(U12(x0, z1))) MARK(U12(length(zeros), x1)) -> c11(A__U12(a__length(a__zeros), x1), MARK(length(zeros))) MARK(U12(length(U11(z0, z1)), x1)) -> c11(A__U12(a__length(a__U11(mark(z0), z1)), x1), MARK(length(U11(z0, z1)))) MARK(U12(length(U12(z0, z1)), x1)) -> c11(A__U12(a__length(a__U12(mark(z0), z1)), x1), MARK(length(U12(z0, z1)))) MARK(U12(length(length(z0)), x1)) -> c11(A__U12(a__length(a__length(mark(z0))), x1), MARK(length(length(z0)))) MARK(U12(length(cons(z0, z1)), x1)) -> c11(A__U12(a__length(cons(mark(z0), z1)), x1), MARK(length(cons(z0, z1)))) MARK(U12(length(s(z0)), x1)) -> c11(A__U12(a__length(s(mark(z0))), x1), MARK(length(s(z0)))) MARK(U12(length(x0), x1)) -> c11(MARK(length(x0))) A__U12(tt, U12(zeros, x1)) -> c4(A__LENGTH(a__U12(a__zeros, x1))) MARK(U11(U12(zeros, x1), x2)) -> c10(A__U11(a__U12(a__zeros, x1), x2)) MARK(U12(U12(zeros, x1), x2)) -> c11(A__U12(a__U12(a__zeros, x1), x2)) MARK(length(U11(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U11(mark(z0), z1), x1)), MARK(U11(U11(z0, z1), x1))) MARK(length(U11(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U11(a__U12(mark(z0), z1), x1)), MARK(U11(U12(z0, z1), x1))) MARK(length(U11(length(z0), x1))) -> c12(A__LENGTH(a__U11(a__length(mark(z0)), x1)), MARK(U11(length(z0), x1))) MARK(length(U11(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U11(cons(mark(z0), z1), x1)), MARK(U11(cons(z0, z1), x1))) MARK(length(U11(tt, x1))) -> c12(A__LENGTH(a__U11(tt, x1)), MARK(U11(tt, x1))) MARK(length(U11(s(z0), x1))) -> c12(A__LENGTH(a__U11(s(mark(z0)), x1)), MARK(U11(s(z0), x1))) MARK(length(U11(x0, z1))) -> c12(MARK(U11(x0, z1))) MARK(length(U11(zeros, x1))) -> c12(A__LENGTH(a__U11(a__zeros, x1))) MARK(length(U12(U11(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U11(mark(z0), z1), x1)), MARK(U12(U11(z0, z1), x1))) MARK(length(U12(U12(z0, z1), x1))) -> c12(A__LENGTH(a__U12(a__U12(mark(z0), z1), x1)), MARK(U12(U12(z0, z1), x1))) MARK(length(U12(length(z0), x1))) -> c12(A__LENGTH(a__U12(a__length(mark(z0)), x1)), MARK(U12(length(z0), x1))) MARK(length(U12(cons(z0, z1), x1))) -> c12(A__LENGTH(a__U12(cons(mark(z0), z1), x1)), MARK(U12(cons(z0, z1), x1))) MARK(length(U12(tt, x1))) -> c12(A__LENGTH(a__U12(tt, x1)), MARK(U12(tt, x1))) MARK(length(U12(s(z0), x1))) -> c12(A__LENGTH(a__U12(s(mark(z0)), x1)), MARK(U12(s(z0), x1))) MARK(length(U12(x0, z1))) -> c12(MARK(U12(x0, z1))) MARK(length(U12(zeros, x1))) -> c12(A__LENGTH(a__U12(a__zeros, x1))) MARK(length(length(zeros))) -> c12(A__LENGTH(a__length(a__zeros)), MARK(length(zeros))) MARK(length(length(U11(z0, z1)))) -> c12(A__LENGTH(a__length(a__U11(mark(z0), z1))), MARK(length(U11(z0, z1)))) MARK(length(length(U12(z0, z1)))) -> c12(A__LENGTH(a__length(a__U12(mark(z0), z1))), MARK(length(U12(z0, z1)))) MARK(length(length(length(z0)))) -> c12(A__LENGTH(a__length(a__length(mark(z0)))), MARK(length(length(z0)))) MARK(length(length(cons(z0, z1)))) -> c12(A__LENGTH(a__length(cons(mark(z0), z1))), MARK(length(cons(z0, z1)))) MARK(length(length(s(z0)))) -> c12(A__LENGTH(a__length(s(mark(z0)))), MARK(length(s(z0)))) MARK(length(length(x0))) -> c12(MARK(length(x0))) MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(A__LENGTH(a__length(nil))) MARK(length(zeros)) -> c12(A__LENGTH(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__zeros, a__U11_2, a__U12_2, a__length_1, mark_1 Defined Pair Symbols: A__U11_2, A__LENGTH_1, MARK_1, A__U12_2 Compound Symbols: c2_1, c7_1, c13_1, c16_1, c4_2, c4_1, c10_1, c11_1, c12_2, c12_1, c10_2, c11_2 ---------------------------------------- (245) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (246) 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) -> a__U12(tt, L) [1] a__U12(tt, L) -> s(a__length(mark(L))) [1] a__length(nil) -> 0 [1] a__length(cons(N, L)) -> a__U11(tt, L) [1] mark(zeros) -> a__zeros [1] mark(U11(X1, X2)) -> a__U11(mark(X1), X2) [1] mark(U12(X1, X2)) -> a__U12(mark(X1), X2) [1] mark(length(X)) -> a__length(mark(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__U12(X1, X2) -> U12(X1, X2) [1] a__length(X) -> length(X) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (247) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (248) 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) -> a__U12(tt, L) [1] a__U12(tt, L) -> s(a__length(mark(L))) [1] a__length(nil) -> 0 [1] a__length(cons(N, L)) -> a__U11(tt, L) [1] mark(zeros) -> a__zeros [1] mark(U11(X1, X2)) -> a__U11(mark(X1), X2) [1] mark(U12(X1, X2)) -> a__U12(mark(X1), X2) [1] mark(length(X)) -> a__length(mark(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__U12(X1, X2) -> U12(X1, X2) [1] a__length(X) -> length(X) [1] The TRS has the following type information: a__zeros :: 0:zeros:cons:tt:s:nil:U11:U12:length cons :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length 0 :: 0:zeros:cons:tt:s:nil:U11:U12:length zeros :: 0:zeros:cons:tt:s:nil:U11:U12:length a__U11 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length tt :: 0:zeros:cons:tt:s:nil:U11:U12:length a__U12 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length s :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length a__length :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length mark :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length nil :: 0:zeros:cons:tt:s:nil:U11:U12:length U11 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length U12 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length length :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length Rewrite Strategy: INNERMOST ---------------------------------------- (249) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: mark_1 a__U11_2 a__zeros a__length_1 a__U12_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (250) 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) -> a__U12(tt, L) [1] a__U12(tt, L) -> s(a__length(mark(L))) [1] a__length(nil) -> 0 [1] a__length(cons(N, L)) -> a__U11(tt, L) [1] mark(zeros) -> a__zeros [1] mark(U11(X1, X2)) -> a__U11(mark(X1), X2) [1] mark(U12(X1, X2)) -> a__U12(mark(X1), X2) [1] mark(length(X)) -> a__length(mark(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__U12(X1, X2) -> U12(X1, X2) [1] a__length(X) -> length(X) [1] The TRS has the following type information: a__zeros :: 0:zeros:cons:tt:s:nil:U11:U12:length cons :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length 0 :: 0:zeros:cons:tt:s:nil:U11:U12:length zeros :: 0:zeros:cons:tt:s:nil:U11:U12:length a__U11 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length tt :: 0:zeros:cons:tt:s:nil:U11:U12:length a__U12 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length s :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length a__length :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length mark :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length nil :: 0:zeros:cons:tt:s:nil:U11:U12:length U11 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length U12 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length length :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length Rewrite Strategy: INNERMOST ---------------------------------------- (251) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (252) 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) -> a__U12(tt, L) [1] a__U12(tt, zeros) -> s(a__length(a__zeros)) [2] a__U12(tt, U11(X1', X2')) -> s(a__length(a__U11(mark(X1'), X2'))) [2] a__U12(tt, U12(X1'', X2'')) -> s(a__length(a__U12(mark(X1''), X2''))) [2] a__U12(tt, length(X')) -> s(a__length(a__length(mark(X')))) [2] a__U12(tt, cons(X11, X21)) -> s(a__length(cons(mark(X11), X21))) [2] a__U12(tt, 0) -> s(a__length(0)) [2] a__U12(tt, tt) -> s(a__length(tt)) [2] a__U12(tt, s(X'')) -> s(a__length(s(mark(X'')))) [2] a__U12(tt, nil) -> s(a__length(nil)) [2] a__length(nil) -> 0 [1] a__length(cons(N, L)) -> a__U11(tt, L) [1] 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(U12(X13, X23), X2)) -> a__U11(a__U12(mark(X13), X23), X2) [2] mark(U11(length(X3), X2)) -> a__U11(a__length(mark(X3)), 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(X4), X2)) -> a__U11(s(mark(X4)), X2) [2] mark(U11(nil, X2)) -> a__U11(nil, X2) [2] mark(U12(zeros, X2)) -> a__U12(a__zeros, X2) [2] mark(U12(U11(X15, X25), X2)) -> a__U12(a__U11(mark(X15), X25), X2) [2] mark(U12(U12(X16, X26), X2)) -> a__U12(a__U12(mark(X16), X26), X2) [2] mark(U12(length(X5), X2)) -> a__U12(a__length(mark(X5)), X2) [2] mark(U12(cons(X17, X27), X2)) -> a__U12(cons(mark(X17), X27), X2) [2] mark(U12(0, X2)) -> a__U12(0, X2) [2] mark(U12(tt, X2)) -> a__U12(tt, X2) [2] mark(U12(s(X6), X2)) -> a__U12(s(mark(X6)), X2) [2] mark(U12(nil, X2)) -> a__U12(nil, X2) [2] mark(length(zeros)) -> a__length(a__zeros) [2] mark(length(U11(X18, X28))) -> a__length(a__U11(mark(X18), X28)) [2] mark(length(U12(X19, X29))) -> a__length(a__U12(mark(X19), X29)) [2] mark(length(length(X7))) -> a__length(a__length(mark(X7))) [2] mark(length(cons(X110, X210))) -> a__length(cons(mark(X110), X210)) [2] mark(length(0)) -> a__length(0) [2] mark(length(tt)) -> a__length(tt) [2] mark(length(s(X8))) -> a__length(s(mark(X8))) [2] mark(length(nil)) -> a__length(nil) [2] 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__U12(X1, X2) -> U12(X1, X2) [1] a__length(X) -> length(X) [1] The TRS has the following type information: a__zeros :: 0:zeros:cons:tt:s:nil:U11:U12:length cons :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length 0 :: 0:zeros:cons:tt:s:nil:U11:U12:length zeros :: 0:zeros:cons:tt:s:nil:U11:U12:length a__U11 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length tt :: 0:zeros:cons:tt:s:nil:U11:U12:length a__U12 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length s :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length a__length :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length mark :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length nil :: 0:zeros:cons:tt:s:nil:U11:U12:length U11 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length U12 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length length :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length Rewrite Strategy: INNERMOST ---------------------------------------- (253) 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 ---------------------------------------- (254) Obligation: Complexity RNTS consisting of the following rules: a__U11(z, z') -{ 1 }-> a__U12(2, L) :|: z = 2, L >= 0, z' = L a__U11(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__U12(z, z') -{ 2 }-> 1 + a__length(a__zeros) :|: z = 2, z' = 3 a__U12(z, z') -{ 2 }-> 1 + a__length(a__length(mark(X'))) :|: z = 2, X' >= 0, z' = 1 + X' a__U12(z, z') -{ 2 }-> 1 + a__length(a__U12(mark(X1''), X2'')) :|: z = 2, z' = 1 + X1'' + X2'', X1'' >= 0, X2'' >= 0 a__U12(z, z') -{ 2 }-> 1 + a__length(a__U11(mark(X1'), X2')) :|: z = 2, z' = 1 + X1' + X2', X2' >= 0, X1' >= 0 a__U12(z, z') -{ 2 }-> 1 + a__length(2) :|: z = 2, z' = 2 a__U12(z, z') -{ 2 }-> 1 + a__length(1) :|: z = 2, z' = 1 a__U12(z, z') -{ 2 }-> 1 + a__length(0) :|: z = 2, z' = 0 a__U12(z, z') -{ 2 }-> 1 + a__length(1 + mark(X'')) :|: z = 2, z' = 1 + X'', X'' >= 0 a__U12(z, z') -{ 2 }-> 1 + a__length(1 + mark(X11) + X21) :|: z = 2, z' = 1 + X11 + X21, X11 >= 0, X21 >= 0 a__U12(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__length(z) -{ 1 }-> a__U11(2, 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(X7))) :|: X7 >= 0, z = 1 + (1 + X7) mark(z) -{ 2 }-> a__length(a__U12(mark(X19), X29)) :|: X29 >= 0, z = 1 + (1 + X19 + X29), X19 >= 0 mark(z) -{ 2 }-> a__length(a__U11(mark(X18), X28)) :|: X18 >= 0, X28 >= 0, z = 1 + (1 + X18 + X28) 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(X8)) :|: X8 >= 0, z = 1 + (1 + X8) mark(z) -{ 2 }-> a__length(1 + mark(X110) + X210) :|: X110 >= 0, z = 1 + (1 + X110 + X210), X210 >= 0 mark(z) -{ 2 }-> a__U12(a__zeros, X2) :|: X2 >= 0, z = 1 + 3 + X2 mark(z) -{ 2 }-> a__U12(a__length(mark(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U12(a__U12(mark(X16), X26), X2) :|: X16 >= 0, X26 >= 0, z = 1 + (1 + X16 + X26) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U12(a__U11(mark(X15), X25), X2) :|: z = 1 + (1 + X15 + X25) + X2, X2 >= 0, X15 >= 0, X25 >= 0 mark(z) -{ 2 }-> a__U12(2, X2) :|: z = 1 + 2 + X2, X2 >= 0 mark(z) -{ 2 }-> a__U12(1, X2) :|: z = 1 + 1 + X2, X2 >= 0 mark(z) -{ 2 }-> a__U12(0, X2) :|: z = 1 + 0 + X2, X2 >= 0 mark(z) -{ 2 }-> a__U12(1 + mark(X6), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U12(1 + mark(X17) + X27, X2) :|: X17 >= 0, X27 >= 0, z = 1 + (1 + X17 + X27) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__zeros, X2) :|: X2 >= 0, z = 1 + 3 + X2 mark(z) -{ 2 }-> a__U11(a__length(mark(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__U12(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(X4), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2 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 ---------------------------------------- (255) 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 :|: ---------------------------------------- (256) Obligation: Complexity RNTS consisting of the following rules: a__U11(z, z') -{ 1 }-> a__U12(2, L) :|: z = 2, L >= 0, z' = L a__U11(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__U12(z, z') -{ 2 }-> 1 + a__length(a__length(mark(X'))) :|: z = 2, X' >= 0, z' = 1 + X' a__U12(z, z') -{ 2 }-> 1 + a__length(a__U12(mark(X1''), X2'')) :|: z = 2, z' = 1 + X1'' + X2'', X1'' >= 0, X2'' >= 0 a__U12(z, z') -{ 2 }-> 1 + a__length(a__U11(mark(X1'), X2')) :|: z = 2, z' = 1 + X1' + X2', X2' >= 0, X1' >= 0 a__U12(z, z') -{ 3 }-> 1 + a__length(3) :|: z = 2, z' = 3 a__U12(z, z') -{ 2 }-> 1 + a__length(2) :|: z = 2, z' = 2 a__U12(z, z') -{ 2 }-> 1 + a__length(1) :|: z = 2, z' = 1 a__U12(z, z') -{ 2 }-> 1 + a__length(0) :|: z = 2, z' = 0 a__U12(z, z') -{ 2 }-> 1 + a__length(1 + mark(X'')) :|: z = 2, z' = 1 + X'', X'' >= 0 a__U12(z, z') -{ 2 }-> 1 + a__length(1 + mark(X11) + X21) :|: z = 2, z' = 1 + X11 + X21, X11 >= 0, X21 >= 0 a__U12(z, z') -{ 3 }-> 1 + a__length(1 + 0 + 3) :|: z = 2, z' = 3 a__U12(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__length(z) -{ 1 }-> a__U11(2, 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(X7))) :|: X7 >= 0, z = 1 + (1 + X7) mark(z) -{ 2 }-> a__length(a__U12(mark(X19), X29)) :|: X29 >= 0, z = 1 + (1 + X19 + X29), X19 >= 0 mark(z) -{ 2 }-> a__length(a__U11(mark(X18), X28)) :|: X18 >= 0, X28 >= 0, z = 1 + (1 + X18 + X28) 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(X8)) :|: X8 >= 0, z = 1 + (1 + X8) mark(z) -{ 2 }-> a__length(1 + mark(X110) + X210) :|: X110 >= 0, z = 1 + (1 + X110 + X210), X210 >= 0 mark(z) -{ 3 }-> a__length(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__U12(a__length(mark(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U12(a__U12(mark(X16), X26), X2) :|: X16 >= 0, X26 >= 0, z = 1 + (1 + X16 + X26) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U12(a__U11(mark(X15), X25), X2) :|: z = 1 + (1 + X15 + X25) + X2, X2 >= 0, X15 >= 0, X25 >= 0 mark(z) -{ 3 }-> a__U12(3, X2) :|: X2 >= 0, z = 1 + 3 + X2 mark(z) -{ 2 }-> a__U12(2, X2) :|: z = 1 + 2 + X2, X2 >= 0 mark(z) -{ 2 }-> a__U12(1, X2) :|: z = 1 + 1 + X2, X2 >= 0 mark(z) -{ 2 }-> a__U12(0, X2) :|: z = 1 + 0 + X2, X2 >= 0 mark(z) -{ 2 }-> a__U12(1 + mark(X6), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U12(1 + mark(X17) + X27, X2) :|: X17 >= 0, X27 >= 0, z = 1 + (1 + X17 + X27) + X2, X2 >= 0 mark(z) -{ 3 }-> a__U12(1 + 0 + 3, X2) :|: X2 >= 0, z = 1 + 3 + X2 mark(z) -{ 2 }-> a__U11(a__length(mark(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__U12(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(X4), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2 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 ---------------------------------------- (257) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (258) Obligation: Complexity RNTS consisting of the following rules: a__U11(z, z') -{ 1 }-> a__U12(2, z') :|: z = 2, z' >= 0 a__U11(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 a__U12(z, z') -{ 2 }-> 1 + a__length(a__length(mark(z' - 1))) :|: z = 2, z' - 1 >= 0 a__U12(z, z') -{ 2 }-> 1 + a__length(a__U12(mark(X1''), X2'')) :|: z = 2, z' = 1 + X1'' + X2'', X1'' >= 0, X2'' >= 0 a__U12(z, z') -{ 2 }-> 1 + a__length(a__U11(mark(X1'), X2')) :|: z = 2, z' = 1 + X1' + X2', X2' >= 0, X1' >= 0 a__U12(z, z') -{ 3 }-> 1 + a__length(3) :|: z = 2, z' = 3 a__U12(z, z') -{ 2 }-> 1 + a__length(2) :|: z = 2, z' = 2 a__U12(z, z') -{ 2 }-> 1 + a__length(1) :|: z = 2, z' = 1 a__U12(z, z') -{ 2 }-> 1 + a__length(0) :|: z = 2, z' = 0 a__U12(z, z') -{ 2 }-> 1 + a__length(1 + mark(z' - 1)) :|: z = 2, z' - 1 >= 0 a__U12(z, z') -{ 2 }-> 1 + a__length(1 + mark(X11) + X21) :|: z = 2, z' = 1 + X11 + X21, X11 >= 0, X21 >= 0 a__U12(z, z') -{ 3 }-> 1 + a__length(1 + 0 + 3) :|: z = 2, z' = 3 a__U12(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 a__length(z) -{ 1 }-> a__U11(2, 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__U12(mark(X19), X29)) :|: X29 >= 0, z = 1 + (1 + X19 + X29), X19 >= 0 mark(z) -{ 2 }-> a__length(a__U11(mark(X18), X28)) :|: X18 >= 0, X28 >= 0, z = 1 + (1 + X18 + X28) 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) + X210) :|: X110 >= 0, z = 1 + (1 + X110 + X210), X210 >= 0 mark(z) -{ 3 }-> a__length(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__U12(a__length(mark(X5)), X2) :|: X5 >= 0, z = 1 + (1 + X5) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U12(a__U12(mark(X16), X26), X2) :|: X16 >= 0, X26 >= 0, z = 1 + (1 + X16 + X26) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U12(a__U11(mark(X15), X25), X2) :|: z = 1 + (1 + X15 + X25) + X2, X2 >= 0, X15 >= 0, X25 >= 0 mark(z) -{ 3 }-> a__U12(3, z - 4) :|: z - 4 >= 0 mark(z) -{ 2 }-> a__U12(2, z - 3) :|: z - 3 >= 0 mark(z) -{ 2 }-> a__U12(1, z - 2) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__U12(0, z - 1) :|: z - 1 >= 0 mark(z) -{ 2 }-> a__U12(1 + mark(X6), X2) :|: X6 >= 0, z = 1 + (1 + X6) + X2, X2 >= 0 mark(z) -{ 2 }-> a__U12(1 + mark(X17) + X27, X2) :|: X17 >= 0, X27 >= 0, z = 1 + (1 + X17 + X27) + X2, X2 >= 0 mark(z) -{ 3 }-> a__U12(1 + 0 + 3, z - 4) :|: z - 4 >= 0 mark(z) -{ 2 }-> a__U11(a__length(mark(X3)), X2) :|: z = 1 + (1 + X3) + X2, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__U11(a__U12(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(X4), X2) :|: X4 >= 0, X2 >= 0, z = 1 + (1 + X4) + X2 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 ---------------------------------------- (259) 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 ---------------------------------------- (260) 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) -> a__U12(tt, L) [1] a__U12(tt, L) -> s(a__length(mark(L))) [1] a__length(nil) -> 0 [1] a__length(cons(N, L)) -> a__U11(tt, L) [1] mark(zeros) -> a__zeros [1] mark(U11(X1, X2)) -> a__U11(mark(X1), X2) [1] mark(U12(X1, X2)) -> a__U12(mark(X1), X2) [1] mark(length(X)) -> a__length(mark(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__U12(X1, X2) -> U12(X1, X2) [1] a__length(X) -> length(X) [1] The TRS has the following type information: a__zeros :: 0:zeros:cons:tt:s:nil:U11:U12:length cons :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length 0 :: 0:zeros:cons:tt:s:nil:U11:U12:length zeros :: 0:zeros:cons:tt:s:nil:U11:U12:length a__U11 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length tt :: 0:zeros:cons:tt:s:nil:U11:U12:length a__U12 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length s :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length a__length :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length mark :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length nil :: 0:zeros:cons:tt:s:nil:U11:U12:length U11 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length U12 :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length length :: 0:zeros:cons:tt:s:nil:U11:U12:length -> 0:zeros:cons:tt:s:nil:U11:U12:length Rewrite Strategy: INNERMOST ---------------------------------------- (261) 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 ---------------------------------------- (262) Obligation: Complexity RNTS consisting of the following rules: a__U11(z, z') -{ 1 }-> a__U12(2, L) :|: z = 2, L >= 0, z' = L a__U11(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__U12(z, z') -{ 1 }-> 1 + a__length(mark(L)) :|: z = 2, L >= 0, z' = L a__U12(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__length(z) -{ 1 }-> a__U11(2, 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__U12(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.