KILLED proof of input_0wGrAjWGCR.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) CompletionProof [UPPER BOUND(ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxTypedWeightedCompleteTrs (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 6 ms] (18) CpxRNTS (19) InliningProof [UPPER BOUND(ID), 284 ms] (20) CpxRNTS (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (24) CdtProblem (25) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms] (46) CdtProblem (47) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 9 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (72) CdtProblem (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtLeafRemovalProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 27 ms] (120) CdtProblem (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 40 ms] (124) CdtProblem (125) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 135 ms] (128) CdtProblem (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 99 ms] (132) CdtProblem (133) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 162 ms] (136) CdtProblem (137) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 165 ms] (140) CdtProblem (141) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 167 ms] (144) CdtProblem (145) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 186 ms] (148) CdtProblem (149) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 234 ms] (152) CdtProblem (153) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 289 ms] (156) CdtProblem (157) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 280 ms] (160) CdtProblem (161) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 326 ms] (164) CdtProblem (165) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 270 ms] (168) CdtProblem (169) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 327 ms] (172) CdtProblem (173) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 392 ms] (176) CdtProblem (177) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 321 ms] (180) CdtProblem (181) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 415 ms] (184) CdtProblem (185) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 451 ms] (188) CdtProblem (189) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 493 ms] (192) CdtProblem (193) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 453 ms] (196) CdtProblem (197) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 469 ms] (200) CdtProblem (201) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 510 ms] (204) CdtProblem (205) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 580 ms] (208) CdtProblem (209) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 520 ms] (212) CdtProblem (213) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 543 ms] (216) CdtProblem (217) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (218) CdtProblem (219) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 641 ms] (220) CdtProblem (221) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) a__U11(tt, L) -> 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) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a__zeros -> cons(0, zeros) [1] a__U11(tt, L) -> 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 ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a__zeros -> cons(0, zeros) [1] a__U11(tt, L) -> 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 ---------------------------------------- (9) 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 ---------------------------------------- (10) 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 ---------------------------------------- (11) 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 ---------------------------------------- (12) 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. ---------------------------------------- (13) 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 ---------------------------------------- (14) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: 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 ---------------------------------------- (15) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (16) 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 ---------------------------------------- (17) 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 ---------------------------------------- (18) 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 ---------------------------------------- (19) 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 :|: ---------------------------------------- (20) 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 ---------------------------------------- (21) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (22) 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 ---------------------------------------- (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__LENGTH(nil) -> c6 A__U12(z0, z1) -> c5 MARK(0) -> c14 MARK(tt) -> c15 A__ZEROS -> c MARK(zeros) -> c9(A__ZEROS) MARK(nil) -> c17 A__U11(z0, z1) -> c3 A__ZEROS -> c1 A__LENGTH(z0) -> c8 ---------------------------------------- (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) 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)) ---------------------------------------- (28) 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 ---------------------------------------- (29) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: A__U12(tt, nil) -> c4(A__LENGTH(nil), MARK(nil)) A__U12(tt, tt) -> c4(A__LENGTH(tt), MARK(tt)) A__U12(tt, 0) -> c4(A__LENGTH(0), MARK(0)) ---------------------------------------- (30) 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 ---------------------------------------- (31) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> 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 ---------------------------------------- (33) 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)) ---------------------------------------- (34) 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 ---------------------------------------- (35) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(U11(nil, x1)) -> c10(A__U11(nil, x1), MARK(nil)) MARK(U11(0, x1)) -> c10(A__U11(0, x1), MARK(0)) ---------------------------------------- (36) 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 ---------------------------------------- (37) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (38) 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 ---------------------------------------- (39) 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)) ---------------------------------------- (40) 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 ---------------------------------------- (41) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(U12(nil, x1)) -> c11(A__U12(nil, x1), MARK(nil)) MARK(U12(0, x1)) -> c11(A__U12(0, x1), MARK(0)) ---------------------------------------- (42) 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 ---------------------------------------- (43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (44) 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 ---------------------------------------- (45) 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)) ---------------------------------------- (46) 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 ---------------------------------------- (47) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(length(0)) -> c12(A__LENGTH(0), MARK(0)) MARK(length(nil)) -> c12(A__LENGTH(nil), MARK(nil)) MARK(length(tt)) -> c12(A__LENGTH(tt), MARK(tt)) ---------------------------------------- (48) 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 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (51) 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))) ---------------------------------------- (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(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 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 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(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 ---------------------------------------- (55) 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))) ---------------------------------------- (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(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 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (59) 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))) ---------------------------------------- (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(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 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> 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 ---------------------------------------- (63) 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)) ---------------------------------------- (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(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 ---------------------------------------- (65) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U12(tt, zeros) -> c4(A__LENGTH(zeros)) ---------------------------------------- (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(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 ---------------------------------------- (67) 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))) ---------------------------------------- (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, 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 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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, 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 ---------------------------------------- (71) 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))) ---------------------------------------- (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, 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 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: a__zeros -> cons(0, zeros) a__zeros -> zeros a__U11(tt, z0) -> 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 ---------------------------------------- (75) 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))) ---------------------------------------- (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, 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 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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, 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 ---------------------------------------- (79) 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)) ---------------------------------------- (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, 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 ---------------------------------------- (81) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(U11(zeros, x0)) -> c10(A__U11(cons(0, zeros), x0)) MARK(U11(zeros, x0)) -> c10(A__U11(zeros, x0)) ---------------------------------------- (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, 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 ---------------------------------------- (83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 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, 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 ---------------------------------------- (85) 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))) ---------------------------------------- (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(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 ---------------------------------------- (87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (89) 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))) ---------------------------------------- (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(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 ---------------------------------------- (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(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 ---------------------------------------- (93) 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))) ---------------------------------------- (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(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 ---------------------------------------- (95) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 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(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 ---------------------------------------- (97) 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)) ---------------------------------------- (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(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 ---------------------------------------- (99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(U12(zeros, x0)) -> c11(A__U12(cons(0, zeros), x0)) MARK(U12(zeros, x0)) -> c11(A__U12(zeros, x0)) ---------------------------------------- (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(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), 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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 ---------------------------------------- (101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (103) 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))) ---------------------------------------- (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(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), 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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))) 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(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 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 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(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 ---------------------------------------- (107) 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))) ---------------------------------------- (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(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 ---------------------------------------- (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(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))) 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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))) 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(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 ---------------------------------------- (111) 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))) ---------------------------------------- (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(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)) -> 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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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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 ---------------------------------------- (113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 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(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)) -> 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)) -> 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MARK(length(length(tt))) -> c12(A__LENGTH(a__length(tt))) MARK(length(length(nil))) -> c12(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, 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), 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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), 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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 ---------------------------------------- (115) 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)) ---------------------------------------- (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(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))) MARK(length(zeros)) -> c12(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(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))) 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 ---------------------------------------- (117) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(length(zeros)) -> c12(A__LENGTH(zeros)) ---------------------------------------- (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(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 ---------------------------------------- (119) 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))) ---------------------------------------- (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(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 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U12(tt, U11(0, z1)) -> c4(A__LENGTH(U11(0, z1))) ---------------------------------------- (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(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))) 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 ---------------------------------------- (123) 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))) ---------------------------------------- (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(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))) 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))) 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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, 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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 ---------------------------------------- (125) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U12(tt, U11(nil, z1)) -> c4(A__LENGTH(U11(nil, z1))) ---------------------------------------- (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(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))) 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(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 ---------------------------------------- (127) 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))) ---------------------------------------- (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(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 ---------------------------------------- (129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U12(tt, U12(0, z1)) -> c4(A__LENGTH(U12(0, z1))) ---------------------------------------- (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(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 ---------------------------------------- (131) 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))) ---------------------------------------- (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(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)) -> 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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)))) 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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))) 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 ---------------------------------------- (133) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U12(tt, U12(nil, z1)) -> c4(A__LENGTH(U12(nil, z1))) ---------------------------------------- (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(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))) 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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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(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 ---------------------------------------- (135) 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))) ---------------------------------------- (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(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))) A__U12(tt, length(0)) -> c4(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, 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, 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 ---------------------------------------- (137) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U12(tt, length(0)) -> c4(A__LENGTH(length(0))) ---------------------------------------- (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, 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 ---------------------------------------- (139) 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))) ---------------------------------------- (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, 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 ---------------------------------------- (141) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__U12(tt, length(tt)) -> c4(A__LENGTH(length(tt))) ---------------------------------------- (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, 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))) 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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 ---------------------------------------- (143) 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))) ---------------------------------------- (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, 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 ---------------------------------------- (145) 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)) ---------------------------------------- (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, 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 ---------------------------------------- (147) 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)) ---------------------------------------- (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, 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))) -> 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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 ---------------------------------------- (149) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U11(U11(0, z1), x1)) -> c10(A__U11(U11(0, z1), x1)) ---------------------------------------- (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, 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), 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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 ---------------------------------------- (151) 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)) ---------------------------------------- (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, 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))) -> 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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))) -> 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 ---------------------------------------- (153) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U11(U11(nil, z1), x1)) -> c10(A__U11(U11(nil, z1), x1)) ---------------------------------------- (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, 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 ---------------------------------------- (155) 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)) ---------------------------------------- (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))) -> 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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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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 ---------------------------------------- (157) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U11(U12(0, z1), x1)) -> c10(A__U11(U12(0, z1), x1)) ---------------------------------------- (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, 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))) 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(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))) 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) 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)) ---------------------------------------- (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, 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 ---------------------------------------- (161) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U11(U12(nil, z1), x1)) -> c10(A__U11(U12(nil, z1), x1)) ---------------------------------------- (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, 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 ---------------------------------------- (163) 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)) ---------------------------------------- (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, 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))) 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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 ---------------------------------------- (165) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U11(length(0), x0)) -> c10(A__U11(length(0), x0)) ---------------------------------------- (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(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))) 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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 ---------------------------------------- (167) 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)) ---------------------------------------- (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(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 ---------------------------------------- (169) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U11(length(tt), x0)) -> c10(A__U11(length(tt), x0)) ---------------------------------------- (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(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 ---------------------------------------- (171) 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)) ---------------------------------------- (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(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), 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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), 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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 ---------------------------------------- (173) 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)) ---------------------------------------- (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(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))) -> 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(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 ---------------------------------------- (175) 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)) ---------------------------------------- (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(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 ---------------------------------------- (177) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U12(U11(0, z1), x1)) -> c11(A__U12(U11(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(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, 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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 ---------------------------------------- (179) 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)) ---------------------------------------- (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(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))) MARK(U12(U11(nil, z1), x1)) -> c11(A__U12(U11(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(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(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 ---------------------------------------- (181) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U12(U11(nil, z1), x1)) -> c11(A__U12(U11(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))) 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 ---------------------------------------- (183) 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)) ---------------------------------------- (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), 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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 ---------------------------------------- (185) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U12(U12(0, z1), x1)) -> c11(A__U12(U12(0, z1), x1)) ---------------------------------------- (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))) 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))) 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(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 ---------------------------------------- (187) 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)) ---------------------------------------- (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))) 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 ---------------------------------------- (189) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U12(U12(nil, z1), x1)) -> c11(A__U12(U12(nil, z1), x1)) ---------------------------------------- (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))) 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))) 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))) 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) 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)) ---------------------------------------- (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))) 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))) 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), 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))) 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 ---------------------------------------- (193) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U12(length(0), x0)) -> c11(A__U12(length(0), 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)) -> 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 ---------------------------------------- (195) 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)) ---------------------------------------- (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))) 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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 ---------------------------------------- (197) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(U12(length(tt), x0)) -> c11(A__U12(length(tt), x0)) ---------------------------------------- (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(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 ---------------------------------------- (199) 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)) ---------------------------------------- (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, 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 ---------------------------------------- (201) 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)) ---------------------------------------- (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(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))) 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))) 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) 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))) ---------------------------------------- (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(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 ---------------------------------------- (205) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(length(U11(0, z1))) -> c12(A__LENGTH(U11(0, z1))) ---------------------------------------- (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))) -> 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), 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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 ---------------------------------------- (207) 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))) ---------------------------------------- (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(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))) MARK(length(U11(nil, z1))) -> c12(A__LENGTH(U11(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(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 ---------------------------------------- (209) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(length(U11(nil, z1))) -> c12(A__LENGTH(U11(nil, z1))) ---------------------------------------- (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))) 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 ---------------------------------------- (211) 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))) ---------------------------------------- (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))) 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)) -> 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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))) 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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), 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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))) 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 ---------------------------------------- (213) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(length(U12(0, z1))) -> c12(A__LENGTH(U12(0, z1))) ---------------------------------------- (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))) 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 ---------------------------------------- (215) 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))) ---------------------------------------- (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))) -> 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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 ---------------------------------------- (217) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(length(U12(nil, z1))) -> c12(A__LENGTH(U12(nil, z1))) ---------------------------------------- (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), 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))) 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))) 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) 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))) ---------------------------------------- (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))) 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 ---------------------------------------- (221) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(length(length(0))) -> c12(A__LENGTH(length(0))) ---------------------------------------- (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), 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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