KILLED proof of input_kLsy2jNcpN.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (6) CdtProblem (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxRelTRS (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxRelTRS (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (12) typed CpxTrs (13) OrderProof [LOWER BOUND(ID), 4 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 9581 ms] (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 2313 ms] (18) BEST (19) proven lower bound (20) LowerBoundPropagationProof [FINISHED, 0 ms] (21) BOUNDS(n^1, INF) (22) typed CpxTrs (23) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (24) CdtProblem (25) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CpxRelTRS (31) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (32) CpxTRS (33) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CpxWeightedTrs (35) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CpxTypedWeightedTrs (37) CompletionProof [UPPER BOUND(ID), 0 ms] (38) CpxTypedWeightedCompleteTrs (39) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CpxTypedWeightedCompleteTrs (41) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 12 ms] (42) CpxRNTS (43) InliningProof [UPPER BOUND(ID), 784 ms] (44) CpxRNTS (45) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CpxRNTS (47) CompletionProof [UPPER BOUND(ID), 0 ms] (48) CpxTypedWeightedCompleteTrs (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms] (64) CdtProblem (65) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 20 ms] (168) CdtProblem (169) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 98 ms] (176) CdtProblem (177) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 84 ms] (180) CdtProblem (181) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 83 ms] (184) CdtProblem (185) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 2 ms] (186) CdtProblem (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 103 ms] (188) CdtProblem (189) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 180 ms] (192) CdtProblem (193) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 183 ms] (196) CdtProblem (197) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 323 ms] (200) CdtProblem (201) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 317 ms] (204) CdtProblem (205) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 354 ms] (208) CdtProblem (209) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 401 ms] (212) CdtProblem (213) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 485 ms] (216) CdtProblem (217) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (218) CdtProblem (219) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 438 ms] (220) CdtProblem (221) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CdtProblem (223) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 485 ms] (224) CdtProblem (225) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (226) CdtProblem (227) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 526 ms] (228) CdtProblem (229) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (230) CdtProblem (231) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 570 ms] (232) CdtProblem (233) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (234) CdtProblem (235) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 587 ms] (236) CdtProblem (237) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (238) CdtProblem (239) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 685 ms] (240) CdtProblem (241) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (242) CdtProblem (243) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (244) CpxWeightedTrs (245) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (246) CpxTypedWeightedTrs (247) CompletionProof [UPPER BOUND(ID), 0 ms] (248) CpxTypedWeightedCompleteTrs (249) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 2 ms] (250) CpxRNTS (251) CompletionProof [UPPER BOUND(ID), 0 ms] (252) CpxTypedWeightedCompleteTrs (253) NarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (254) CpxTypedWeightedCompleteTrs (255) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (256) CpxRNTS (257) InliningProof [UPPER BOUND(ID), 332 ms] (258) CpxRNTS (259) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (260) CpxRNTS ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a__incr(nil) -> nil a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L)) a__adx(nil) -> nil a__adx(cons(X, L)) -> a__incr(cons(mark(X), adx(L))) a__nats -> a__adx(a__zeros) a__zeros -> cons(0, zeros) a__head(cons(X, L)) -> mark(X) a__tail(cons(X, L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil) -> nil mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats -> nats a__zeros -> zeros a__head(X) -> head(X) a__tail(X) -> tail(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__incr(nil) -> nil a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L)) a__adx(nil) -> nil a__adx(cons(X, L)) -> a__incr(cons(mark(X), adx(L))) a__nats -> a__adx(a__zeros) a__zeros -> cons(0', zeros) a__head(cons(X, L)) -> mark(X) a__tail(cons(X, L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil) -> nil mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0') -> 0' a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats -> nats a__zeros -> zeros a__head(X) -> head(X) a__tail(X) -> tail(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__incr(nil) -> nil a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L)) a__adx(nil) -> nil a__adx(cons(X, L)) -> a__incr(cons(mark(X), adx(L))) a__nats -> a__adx(a__zeros) a__zeros -> cons(0, zeros) a__head(cons(X, L)) -> mark(X) a__tail(cons(X, L)) -> mark(L) mark(incr(X)) -> a__incr(mark(X)) mark(adx(X)) -> a__adx(mark(X)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(nil) -> nil mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(0) -> 0 a__incr(X) -> incr(X) a__adx(X) -> adx(X) a__nats -> nats a__zeros -> zeros a__head(X) -> head(X) a__tail(X) -> tail(X) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(nil) -> c A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__INCR(z0) -> c2 A__ADX(nil) -> c3 A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__ADX(z0) -> c5 A__NATS -> c6(A__ADX(a__zeros), A__ZEROS) A__NATS -> c7 A__ZEROS -> c8 A__ZEROS -> c9 A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__HEAD(z0) -> c11 A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) A__TAIL(z0) -> c13 MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(zeros) -> c17(A__ZEROS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(nil) -> c20 MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(0) -> c23 S tuples: A__INCR(nil) -> c A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__INCR(z0) -> c2 A__ADX(nil) -> c3 A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__ADX(z0) -> c5 A__NATS -> c6(A__ADX(a__zeros), A__ZEROS) A__NATS -> c7 A__ZEROS -> c8 A__ZEROS -> c9 A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__HEAD(z0) -> c11 A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) A__TAIL(z0) -> c13 MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(zeros) -> c17(A__ZEROS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(nil) -> c20 MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(0) -> c23 K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__NATS, A__ZEROS, A__HEAD_1, A__TAIL_1, MARK_1 Compound Symbols: c, c1_1, c2, c3, c4_2, c5, c6_2, c7, c8, c9, c10_1, c11, c12_1, c13, c14_2, c15_2, c16_1, c17_1, c18_2, c19_2, c20, c21_1, c22_1, c23 ---------------------------------------- (7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__INCR(nil) -> c A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__INCR(z0) -> c2 A__ADX(nil) -> c3 A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__ADX(z0) -> c5 A__NATS -> c6(A__ADX(a__zeros), A__ZEROS) A__NATS -> c7 A__ZEROS -> c8 A__ZEROS -> c9 A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__HEAD(z0) -> c11 A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) A__TAIL(z0) -> c13 MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(zeros) -> c17(A__ZEROS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(nil) -> c20 MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(0) -> c23 The (relative) TRS S consists of the following rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (9) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (10) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: A__INCR(nil) -> c A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__INCR(z0) -> c2 A__ADX(nil) -> c3 A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__ADX(z0) -> c5 A__NATS -> c6(A__ADX(a__zeros), A__ZEROS) A__NATS -> c7 A__ZEROS -> c8 A__ZEROS -> c9 A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__HEAD(z0) -> c11 A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) A__TAIL(z0) -> c13 MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(zeros) -> c17(A__ZEROS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(nil) -> c20 MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(0') -> c23 The (relative) TRS S consists of the following rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0', zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0') -> 0' Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (12) Obligation: Innermost TRS: Rules: A__INCR(nil) -> c A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__INCR(z0) -> c2 A__ADX(nil) -> c3 A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__ADX(z0) -> c5 A__NATS -> c6(A__ADX(a__zeros), A__ZEROS) A__NATS -> c7 A__ZEROS -> c8 A__ZEROS -> c9 A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__HEAD(z0) -> c11 A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) A__TAIL(z0) -> c13 MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(zeros) -> c17(A__ZEROS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(nil) -> c20 MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(0') -> c23 a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0', zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0') -> 0' Types: A__INCR :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c:c1:c2 nil :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c :: c:c1:c2 cons :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c1 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c:c1:c2 MARK :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c2 :: c:c1:c2 A__ADX :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c:c1:c2 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c3:c4:c5 mark :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' adx :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c5 :: c3:c4:c5 A__NATS :: c6:c7 c6 :: c3:c4:c5 -> c8:c9 -> c6:c7 a__zeros :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' A__ZEROS :: c8:c9 c7 :: c6:c7 c8 :: c8:c9 c9 :: c8:c9 A__HEAD :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c10:c11 c10 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c10:c11 c11 :: c10:c11 A__TAIL :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c12:c13 c12 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c12:c13 c13 :: c12:c13 incr :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c14 :: c:c1:c2 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c15 :: c3:c4:c5 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 nats :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c16 :: c6:c7 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 zeros :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c17 :: c8:c9 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 head :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c18 :: c10:c11 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 tail :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c19 :: c12:c13 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c20 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c21 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 s :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c22 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 0' :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c23 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 a__incr :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__adx :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__nats :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__head :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__tail :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' hole_c:c1:c21_24 :: c:c1:c2 hole_nil:cons:adx:incr:nats:zeros:head:tail:s:0'2_24 :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' hole_c14:c15:c16:c17:c18:c19:c20:c21:c22:c233_24 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 hole_c3:c4:c54_24 :: c3:c4:c5 hole_c6:c75_24 :: c6:c7 hole_c8:c96_24 :: c8:c9 hole_c10:c117_24 :: c10:c11 hole_c12:c138_24 :: c12:c13 gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24 :: Nat -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24 :: Nat -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 ---------------------------------------- (13) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: A__INCR, MARK, A__ADX, mark, A__NATS, A__HEAD, A__TAIL, a__incr, a__adx, a__nats, a__head, a__tail They will be analysed ascendingly in the following order: A__INCR = MARK A__INCR = A__ADX A__INCR = A__NATS A__INCR = A__HEAD A__INCR = A__TAIL MARK = A__ADX mark < MARK MARK = A__NATS MARK = A__HEAD MARK = A__TAIL mark < A__ADX A__ADX = A__NATS A__ADX = A__HEAD A__ADX = A__TAIL mark = a__incr mark = a__adx mark = a__nats mark = a__head mark = a__tail A__NATS = A__HEAD A__NATS = A__TAIL A__HEAD = A__TAIL a__incr = a__adx a__incr = a__nats a__incr = a__head a__incr = a__tail a__adx = a__nats a__adx = a__head a__adx = a__tail a__nats = a__head a__nats = a__tail a__head = a__tail ---------------------------------------- (14) Obligation: Innermost TRS: Rules: A__INCR(nil) -> c A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__INCR(z0) -> c2 A__ADX(nil) -> c3 A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__ADX(z0) -> c5 A__NATS -> c6(A__ADX(a__zeros), A__ZEROS) A__NATS -> c7 A__ZEROS -> c8 A__ZEROS -> c9 A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__HEAD(z0) -> c11 A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) A__TAIL(z0) -> c13 MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(zeros) -> c17(A__ZEROS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(nil) -> c20 MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(0') -> c23 a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0', zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0') -> 0' Types: A__INCR :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c:c1:c2 nil :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c :: c:c1:c2 cons :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c1 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c:c1:c2 MARK :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c2 :: c:c1:c2 A__ADX :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c:c1:c2 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c3:c4:c5 mark :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' adx :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c5 :: c3:c4:c5 A__NATS :: c6:c7 c6 :: c3:c4:c5 -> c8:c9 -> c6:c7 a__zeros :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' A__ZEROS :: c8:c9 c7 :: c6:c7 c8 :: c8:c9 c9 :: c8:c9 A__HEAD :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c10:c11 c10 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c10:c11 c11 :: c10:c11 A__TAIL :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c12:c13 c12 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c12:c13 c13 :: c12:c13 incr :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c14 :: c:c1:c2 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c15 :: c3:c4:c5 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 nats :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c16 :: c6:c7 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 zeros :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c17 :: c8:c9 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 head :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c18 :: c10:c11 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 tail :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c19 :: c12:c13 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c20 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c21 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 s :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c22 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 0' :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c23 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 a__incr :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__adx :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__nats :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__head :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__tail :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' hole_c:c1:c21_24 :: c:c1:c2 hole_nil:cons:adx:incr:nats:zeros:head:tail:s:0'2_24 :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' hole_c14:c15:c16:c17:c18:c19:c20:c21:c22:c233_24 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 hole_c3:c4:c54_24 :: c3:c4:c5 hole_c6:c75_24 :: c6:c7 hole_c8:c96_24 :: c8:c9 hole_c10:c117_24 :: c10:c11 hole_c12:c138_24 :: c12:c13 gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24 :: Nat -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24 :: Nat -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 Generator Equations: gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(0) <=> nil gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(+(x, 1)) <=> cons(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(x), nil) gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24(0) <=> c16(c6(c3, c8)) gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24(+(x, 1)) <=> c14(c, gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24(x)) The following defined symbols remain to be analysed: a__incr, A__INCR, MARK, A__ADX, mark, A__NATS, A__HEAD, A__TAIL, a__adx, a__nats, a__head, a__tail They will be analysed ascendingly in the following order: A__INCR = MARK A__INCR = A__ADX A__INCR = A__NATS A__INCR = A__HEAD A__INCR = A__TAIL MARK = A__ADX mark < MARK MARK = A__NATS MARK = A__HEAD MARK = A__TAIL mark < A__ADX A__ADX = A__NATS A__ADX = A__HEAD A__ADX = A__TAIL mark = a__incr mark = a__adx mark = a__nats mark = a__head mark = a__tail A__NATS = A__HEAD A__NATS = A__TAIL A__HEAD = A__TAIL a__incr = a__adx a__incr = a__nats a__incr = a__head a__incr = a__tail a__adx = a__nats a__adx = a__head a__adx = a__tail a__nats = a__head a__nats = a__tail a__head = a__tail ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: mark(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(n101768_24)) -> gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(n101768_24), rt in Omega(0) Induction Base: mark(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(0)) ->_R^Omega(0) nil Induction Step: mark(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(+(n101768_24, 1))) ->_R^Omega(0) cons(mark(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(n101768_24)), nil) ->_IH cons(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(c101769_24), nil) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (16) Obligation: Innermost TRS: Rules: A__INCR(nil) -> c A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__INCR(z0) -> c2 A__ADX(nil) -> c3 A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__ADX(z0) -> c5 A__NATS -> c6(A__ADX(a__zeros), A__ZEROS) A__NATS -> c7 A__ZEROS -> c8 A__ZEROS -> c9 A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__HEAD(z0) -> c11 A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) A__TAIL(z0) -> c13 MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(zeros) -> c17(A__ZEROS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(nil) -> c20 MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(0') -> c23 a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0', zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0') -> 0' Types: A__INCR :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c:c1:c2 nil :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c :: c:c1:c2 cons :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c1 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c:c1:c2 MARK :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c2 :: c:c1:c2 A__ADX :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c:c1:c2 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c3:c4:c5 mark :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' adx :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c5 :: c3:c4:c5 A__NATS :: c6:c7 c6 :: c3:c4:c5 -> c8:c9 -> c6:c7 a__zeros :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' A__ZEROS :: c8:c9 c7 :: c6:c7 c8 :: c8:c9 c9 :: c8:c9 A__HEAD :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c10:c11 c10 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c10:c11 c11 :: c10:c11 A__TAIL :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c12:c13 c12 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c12:c13 c13 :: c12:c13 incr :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c14 :: c:c1:c2 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c15 :: c3:c4:c5 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 nats :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c16 :: c6:c7 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 zeros :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c17 :: c8:c9 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 head :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c18 :: c10:c11 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 tail :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c19 :: c12:c13 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c20 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c21 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 s :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c22 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 0' :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c23 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 a__incr :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__adx :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__nats :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__head :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__tail :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' hole_c:c1:c21_24 :: c:c1:c2 hole_nil:cons:adx:incr:nats:zeros:head:tail:s:0'2_24 :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' hole_c14:c15:c16:c17:c18:c19:c20:c21:c22:c233_24 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 hole_c3:c4:c54_24 :: c3:c4:c5 hole_c6:c75_24 :: c6:c7 hole_c8:c96_24 :: c8:c9 hole_c10:c117_24 :: c10:c11 hole_c12:c138_24 :: c12:c13 gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24 :: Nat -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24 :: Nat -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 Lemmas: mark(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(n101768_24)) -> gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(n101768_24), rt in Omega(0) Generator Equations: gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(0) <=> nil gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(+(x, 1)) <=> cons(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(x), nil) gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24(0) <=> c16(c6(c3, c8)) gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24(+(x, 1)) <=> c14(c, gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24(x)) The following defined symbols remain to be analysed: a__adx, A__INCR, MARK, A__ADX, A__NATS, A__HEAD, A__TAIL, a__incr, a__nats, a__head, a__tail They will be analysed ascendingly in the following order: A__INCR = MARK A__INCR = A__ADX A__INCR = A__NATS A__INCR = A__HEAD A__INCR = A__TAIL MARK = A__ADX mark < MARK MARK = A__NATS MARK = A__HEAD MARK = A__TAIL mark < A__ADX A__ADX = A__NATS A__ADX = A__HEAD A__ADX = A__TAIL mark = a__incr mark = a__adx mark = a__nats mark = a__head mark = a__tail A__NATS = A__HEAD A__NATS = A__TAIL A__HEAD = A__TAIL a__incr = a__adx a__incr = a__nats a__incr = a__head a__incr = a__tail a__adx = a__nats a__adx = a__head a__adx = a__tail a__nats = a__head a__nats = a__tail a__head = a__tail ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: MARK(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(+(1, n120961_24))) -> *11_24, rt in Omega(n120961_24) Induction Base: MARK(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(+(1, 0))) Induction Step: MARK(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(+(1, +(n120961_24, 1)))) ->_R^Omega(1) c21(MARK(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(+(1, n120961_24)))) ->_IH c21(*11_24) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Complex Obligation (BEST) ---------------------------------------- (19) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: A__INCR(nil) -> c A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__INCR(z0) -> c2 A__ADX(nil) -> c3 A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__ADX(z0) -> c5 A__NATS -> c6(A__ADX(a__zeros), A__ZEROS) A__NATS -> c7 A__ZEROS -> c8 A__ZEROS -> c9 A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__HEAD(z0) -> c11 A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) A__TAIL(z0) -> c13 MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(zeros) -> c17(A__ZEROS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(nil) -> c20 MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(0') -> c23 a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0', zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0') -> 0' Types: A__INCR :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c:c1:c2 nil :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c :: c:c1:c2 cons :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c1 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c:c1:c2 MARK :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c2 :: c:c1:c2 A__ADX :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c:c1:c2 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c3:c4:c5 mark :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' adx :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c5 :: c3:c4:c5 A__NATS :: c6:c7 c6 :: c3:c4:c5 -> c8:c9 -> c6:c7 a__zeros :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' A__ZEROS :: c8:c9 c7 :: c6:c7 c8 :: c8:c9 c9 :: c8:c9 A__HEAD :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c10:c11 c10 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c10:c11 c11 :: c10:c11 A__TAIL :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c12:c13 c12 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c12:c13 c13 :: c12:c13 incr :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c14 :: c:c1:c2 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c15 :: c3:c4:c5 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 nats :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c16 :: c6:c7 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 zeros :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c17 :: c8:c9 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 head :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c18 :: c10:c11 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 tail :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c19 :: c12:c13 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c20 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c21 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 s :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c22 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 0' :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c23 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 a__incr :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__adx :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__nats :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__head :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__tail :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' hole_c:c1:c21_24 :: c:c1:c2 hole_nil:cons:adx:incr:nats:zeros:head:tail:s:0'2_24 :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' hole_c14:c15:c16:c17:c18:c19:c20:c21:c22:c233_24 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 hole_c3:c4:c54_24 :: c3:c4:c5 hole_c6:c75_24 :: c6:c7 hole_c8:c96_24 :: c8:c9 hole_c10:c117_24 :: c10:c11 hole_c12:c138_24 :: c12:c13 gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24 :: Nat -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24 :: Nat -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 Lemmas: mark(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(n101768_24)) -> gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(n101768_24), rt in Omega(0) Generator Equations: gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(0) <=> nil gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(+(x, 1)) <=> cons(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(x), nil) gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24(0) <=> c16(c6(c3, c8)) gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24(+(x, 1)) <=> c14(c, gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24(x)) The following defined symbols remain to be analysed: MARK, A__INCR, A__ADX, A__NATS, A__HEAD, A__TAIL They will be analysed ascendingly in the following order: A__INCR = MARK A__INCR = A__ADX A__INCR = A__NATS A__INCR = A__HEAD A__INCR = A__TAIL MARK = A__ADX MARK = A__NATS MARK = A__HEAD MARK = A__TAIL A__ADX = A__NATS A__ADX = A__HEAD A__ADX = A__TAIL A__NATS = A__HEAD A__NATS = A__TAIL A__HEAD = A__TAIL ---------------------------------------- (20) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (21) BOUNDS(n^1, INF) ---------------------------------------- (22) Obligation: Innermost TRS: Rules: A__INCR(nil) -> c A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__INCR(z0) -> c2 A__ADX(nil) -> c3 A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__ADX(z0) -> c5 A__NATS -> c6(A__ADX(a__zeros), A__ZEROS) A__NATS -> c7 A__ZEROS -> c8 A__ZEROS -> c9 A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__HEAD(z0) -> c11 A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) A__TAIL(z0) -> c13 MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(zeros) -> c17(A__ZEROS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(nil) -> c20 MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(0') -> c23 a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0', zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0') -> 0' Types: A__INCR :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c:c1:c2 nil :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c :: c:c1:c2 cons :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c1 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c:c1:c2 MARK :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c2 :: c:c1:c2 A__ADX :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c:c1:c2 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c3:c4:c5 mark :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' adx :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c5 :: c3:c4:c5 A__NATS :: c6:c7 c6 :: c3:c4:c5 -> c8:c9 -> c6:c7 a__zeros :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' A__ZEROS :: c8:c9 c7 :: c6:c7 c8 :: c8:c9 c9 :: c8:c9 A__HEAD :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c10:c11 c10 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c10:c11 c11 :: c10:c11 A__TAIL :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> c12:c13 c12 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c12:c13 c13 :: c12:c13 incr :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c14 :: c:c1:c2 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c15 :: c3:c4:c5 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 nats :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c16 :: c6:c7 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 zeros :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c17 :: c8:c9 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 head :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c18 :: c10:c11 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 tail :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c19 :: c12:c13 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c20 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 c21 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 s :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' c22 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 0' :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' c23 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 a__incr :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__adx :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__nats :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__head :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' a__tail :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' hole_c:c1:c21_24 :: c:c1:c2 hole_nil:cons:adx:incr:nats:zeros:head:tail:s:0'2_24 :: nil:cons:adx:incr:nats:zeros:head:tail:s:0' hole_c14:c15:c16:c17:c18:c19:c20:c21:c22:c233_24 :: c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 hole_c3:c4:c54_24 :: c3:c4:c5 hole_c6:c75_24 :: c6:c7 hole_c8:c96_24 :: c8:c9 hole_c10:c117_24 :: c10:c11 hole_c12:c138_24 :: c12:c13 gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24 :: Nat -> nil:cons:adx:incr:nats:zeros:head:tail:s:0' gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24 :: Nat -> c14:c15:c16:c17:c18:c19:c20:c21:c22:c23 Lemmas: mark(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(n101768_24)) -> gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(n101768_24), rt in Omega(0) MARK(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(+(1, n120961_24))) -> *11_24, rt in Omega(n120961_24) Generator Equations: gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(0) <=> nil gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(+(x, 1)) <=> cons(gen_nil:cons:adx:incr:nats:zeros:head:tail:s:0'9_24(x), nil) gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24(0) <=> c16(c6(c3, c8)) gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24(+(x, 1)) <=> c14(c, gen_c14:c15:c16:c17:c18:c19:c20:c21:c22:c2310_24(x)) The following defined symbols remain to be analysed: A__INCR, A__ADX, A__NATS, A__HEAD, A__TAIL They will be analysed ascendingly in the following order: A__INCR = MARK A__INCR = A__ADX A__INCR = A__NATS A__INCR = A__HEAD A__INCR = A__TAIL MARK = A__ADX MARK = A__NATS MARK = A__HEAD MARK = A__TAIL A__ADX = A__NATS A__ADX = A__HEAD A__ADX = A__TAIL A__NATS = A__HEAD A__NATS = A__TAIL A__HEAD = A__TAIL ---------------------------------------- (23) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(nil) -> c A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__INCR(z0) -> c2 A__ADX(nil) -> c3 A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__ADX(z0) -> c5 A__NATS -> c6(A__ADX(a__zeros), A__ZEROS) A__NATS -> c7 A__ZEROS -> c8 A__ZEROS -> c9 A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__HEAD(z0) -> c11 A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) A__TAIL(z0) -> c13 MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(zeros) -> c17(A__ZEROS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(nil) -> c20 MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(0) -> c23 S tuples: A__INCR(nil) -> c A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__INCR(z0) -> c2 A__ADX(nil) -> c3 A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__ADX(z0) -> c5 A__NATS -> c6(A__ADX(a__zeros), A__ZEROS) A__NATS -> c7 A__ZEROS -> c8 A__ZEROS -> c9 A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__HEAD(z0) -> c11 A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) A__TAIL(z0) -> c13 MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(zeros) -> c17(A__ZEROS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(nil) -> c20 MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(0) -> c23 K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__NATS, A__ZEROS, A__HEAD_1, A__TAIL_1, MARK_1 Compound Symbols: c, c1_1, c2, c3, c4_2, c5, c6_2, c7, c8, c9, c10_1, c11, c12_1, c13, c14_2, c15_2, c16_1, c17_1, c18_2, c19_2, c20, c21_1, c22_1, c23 ---------------------------------------- (25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: A__ADX(nil) -> c3 MARK(0) -> c23 A__ZEROS -> c8 A__HEAD(z0) -> c11 A__NATS -> c7 A__ZEROS -> c9 A__INCR(nil) -> c MARK(nil) -> c20 A__ADX(z0) -> c5 A__TAIL(z0) -> c13 MARK(zeros) -> c17(A__ZEROS) A__INCR(z0) -> c2 ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__NATS -> c6(A__ADX(a__zeros), A__ZEROS) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__NATS -> c6(A__ADX(a__zeros), A__ZEROS) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__NATS, A__HEAD_1, A__TAIL_1, MARK_1 Compound Symbols: c1_1, c4_2, c6_2, c10_1, c12_1, c14_2, c15_2, c16_1, c18_2, c19_2, c21_1, c22_1 ---------------------------------------- (27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c14_2, c15_2, c16_1, c18_2, c19_2, c21_1, c22_1, c6_1 ---------------------------------------- (29) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (30) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) The (relative) TRS S consists of the following rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (31) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (32) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (33) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (34) 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__INCR(cons(z0, z1)) -> c1(MARK(z0)) [1] A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) [1] A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) [1] A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) [1] MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) [1] MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) [1] MARK(nats) -> c16(A__NATS) [1] MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) [1] MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) [1] MARK(cons(z0, z1)) -> c21(MARK(z0)) [1] MARK(s(z0)) -> c22(MARK(z0)) [1] A__NATS -> c6(A__ADX(a__zeros)) [1] a__incr(nil) -> nil [0] a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) [0] a__incr(z0) -> incr(z0) [0] a__adx(nil) -> nil [0] a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) [0] a__adx(z0) -> adx(z0) [0] a__nats -> a__adx(a__zeros) [0] a__nats -> nats [0] a__zeros -> cons(0, zeros) [0] a__zeros -> zeros [0] a__head(cons(z0, z1)) -> mark(z0) [0] a__head(z0) -> head(z0) [0] a__tail(cons(z0, z1)) -> mark(z1) [0] a__tail(z0) -> tail(z0) [0] mark(incr(z0)) -> a__incr(mark(z0)) [0] mark(adx(z0)) -> a__adx(mark(z0)) [0] mark(nats) -> a__nats [0] mark(zeros) -> a__zeros [0] mark(head(z0)) -> a__head(mark(z0)) [0] mark(tail(z0)) -> a__tail(mark(z0)) [0] mark(nil) -> nil [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(s(z0)) -> s(mark(z0)) [0] mark(0) -> 0 [0] Rewrite Strategy: INNERMOST ---------------------------------------- (35) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (36) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) [1] A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) [1] A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) [1] A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) [1] MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) [1] MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) [1] MARK(nats) -> c16(A__NATS) [1] MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) [1] MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) [1] MARK(cons(z0, z1)) -> c21(MARK(z0)) [1] MARK(s(z0)) -> c22(MARK(z0)) [1] A__NATS -> c6(A__ADX(a__zeros)) [1] a__incr(nil) -> nil [0] a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) [0] a__incr(z0) -> incr(z0) [0] a__adx(nil) -> nil [0] a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) [0] a__adx(z0) -> adx(z0) [0] a__nats -> a__adx(a__zeros) [0] a__nats -> nats [0] a__zeros -> cons(0, zeros) [0] a__zeros -> zeros [0] a__head(cons(z0, z1)) -> mark(z0) [0] a__head(z0) -> head(z0) [0] a__tail(cons(z0, z1)) -> mark(z1) [0] a__tail(z0) -> tail(z0) [0] mark(incr(z0)) -> a__incr(mark(z0)) [0] mark(adx(z0)) -> a__adx(mark(z0)) [0] mark(nats) -> a__nats [0] mark(zeros) -> a__zeros [0] mark(head(z0)) -> a__head(mark(z0)) [0] mark(tail(z0)) -> a__tail(mark(z0)) [0] mark(nil) -> nil [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(s(z0)) -> s(mark(z0)) [0] mark(0) -> 0 [0] The TRS has the following type information: A__INCR :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> c1 cons :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> cons:adx:incr:nats:head:tail:s:nil:0:zeros -> cons:adx:incr:nats:head:tail:s:nil:0:zeros c1 :: c14:c15:c16:c18:c19:c21:c22 -> c1 MARK :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> c14:c15:c16:c18:c19:c21:c22 A__ADX :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> c4 c4 :: c1 -> c14:c15:c16:c18:c19:c21:c22 -> c4 mark :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> cons:adx:incr:nats:head:tail:s:nil:0:zeros adx :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> cons:adx:incr:nats:head:tail:s:nil:0:zeros A__HEAD :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> c10 c10 :: c14:c15:c16:c18:c19:c21:c22 -> c10 A__TAIL :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> c12 c12 :: c14:c15:c16:c18:c19:c21:c22 -> c12 incr :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> cons:adx:incr:nats:head:tail:s:nil:0:zeros c14 :: c1 -> c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 c15 :: c4 -> c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 nats :: cons:adx:incr:nats:head:tail:s:nil:0:zeros c16 :: c6 -> c14:c15:c16:c18:c19:c21:c22 A__NATS :: c6 head :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> cons:adx:incr:nats:head:tail:s:nil:0:zeros c18 :: c10 -> c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 tail :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> cons:adx:incr:nats:head:tail:s:nil:0:zeros c19 :: c12 -> c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 c21 :: c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 s :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> cons:adx:incr:nats:head:tail:s:nil:0:zeros c22 :: c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 c6 :: c4 -> c6 a__zeros :: cons:adx:incr:nats:head:tail:s:nil:0:zeros a__incr :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> cons:adx:incr:nats:head:tail:s:nil:0:zeros nil :: cons:adx:incr:nats:head:tail:s:nil:0:zeros a__adx :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> cons:adx:incr:nats:head:tail:s:nil:0:zeros a__nats :: cons:adx:incr:nats:head:tail:s:nil:0:zeros 0 :: cons:adx:incr:nats:head:tail:s:nil:0:zeros zeros :: cons:adx:incr:nats:head:tail:s:nil:0:zeros a__head :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> cons:adx:incr:nats:head:tail:s:nil:0:zeros a__tail :: cons:adx:incr:nats:head:tail:s:nil:0:zeros -> cons:adx:incr:nats:head:tail:s:nil:0:zeros Rewrite Strategy: INNERMOST ---------------------------------------- (37) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: A__INCR_1 A__ADX_1 A__HEAD_1 A__TAIL_1 MARK_1 A__NATS (c) The following functions are completely defined: a__incr_1 a__adx_1 a__nats a__zeros a__head_1 a__tail_1 mark_1 Due to the following rules being added: a__incr(v0) -> null_a__incr [0] a__adx(v0) -> null_a__adx [0] a__nats -> null_a__nats [0] a__zeros -> null_a__zeros [0] a__head(v0) -> null_a__head [0] a__tail(v0) -> null_a__tail [0] mark(v0) -> null_mark [0] And the following fresh constants: null_a__incr, null_a__adx, null_a__nats, null_a__zeros, null_a__head, null_a__tail, null_mark, const, const1, const2, const3, const4, const5 ---------------------------------------- (38) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) [1] A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) [1] A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) [1] A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) [1] MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) [1] MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) [1] MARK(nats) -> c16(A__NATS) [1] MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) [1] MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) [1] MARK(cons(z0, z1)) -> c21(MARK(z0)) [1] MARK(s(z0)) -> c22(MARK(z0)) [1] A__NATS -> c6(A__ADX(a__zeros)) [1] a__incr(nil) -> nil [0] a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) [0] a__incr(z0) -> incr(z0) [0] a__adx(nil) -> nil [0] a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) [0] a__adx(z0) -> adx(z0) [0] a__nats -> a__adx(a__zeros) [0] a__nats -> nats [0] a__zeros -> cons(0, zeros) [0] a__zeros -> zeros [0] a__head(cons(z0, z1)) -> mark(z0) [0] a__head(z0) -> head(z0) [0] a__tail(cons(z0, z1)) -> mark(z1) [0] a__tail(z0) -> tail(z0) [0] mark(incr(z0)) -> a__incr(mark(z0)) [0] mark(adx(z0)) -> a__adx(mark(z0)) [0] mark(nats) -> a__nats [0] mark(zeros) -> a__zeros [0] mark(head(z0)) -> a__head(mark(z0)) [0] mark(tail(z0)) -> a__tail(mark(z0)) [0] mark(nil) -> nil [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(s(z0)) -> s(mark(z0)) [0] mark(0) -> 0 [0] a__incr(v0) -> null_a__incr [0] a__adx(v0) -> null_a__adx [0] a__nats -> null_a__nats [0] a__zeros -> null_a__zeros [0] a__head(v0) -> null_a__head [0] a__tail(v0) -> null_a__tail [0] mark(v0) -> null_mark [0] The TRS has the following type information: A__INCR :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> c1 cons :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c1 :: c14:c15:c16:c18:c19:c21:c22 -> c1 MARK :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> c14:c15:c16:c18:c19:c21:c22 A__ADX :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> c4 c4 :: c1 -> c14:c15:c16:c18:c19:c21:c22 -> c4 mark :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark adx :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark A__HEAD :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> c10 c10 :: c14:c15:c16:c18:c19:c21:c22 -> c10 A__TAIL :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> c12 c12 :: c14:c15:c16:c18:c19:c21:c22 -> c12 incr :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c14 :: c1 -> c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 c15 :: c4 -> c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 nats :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c16 :: c6 -> c14:c15:c16:c18:c19:c21:c22 A__NATS :: c6 head :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c18 :: c10 -> c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 tail :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c19 :: c12 -> c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 c21 :: c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 s :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c22 :: c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 c6 :: c4 -> c6 a__zeros :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark a__incr :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark nil :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark a__adx :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark a__nats :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark 0 :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark zeros :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark a__head :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark a__tail :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__incr :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__adx :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__nats :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__zeros :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__head :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__tail :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_mark :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark const :: c1 const1 :: c14:c15:c16:c18:c19:c21:c22 const2 :: c4 const3 :: c10 const4 :: c12 const5 :: c6 Rewrite Strategy: INNERMOST ---------------------------------------- (39) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (40) 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__INCR(cons(z0, z1)) -> c1(MARK(z0)) [1] A__ADX(cons(incr(z0'), z1)) -> c4(A__INCR(cons(a__incr(mark(z0')), adx(z1))), MARK(incr(z0'))) [1] A__ADX(cons(adx(z0''), z1)) -> c4(A__INCR(cons(a__adx(mark(z0'')), adx(z1))), MARK(adx(z0''))) [1] A__ADX(cons(nats, z1)) -> c4(A__INCR(cons(a__nats, adx(z1))), MARK(nats)) [1] A__ADX(cons(zeros, z1)) -> c4(A__INCR(cons(a__zeros, adx(z1))), MARK(zeros)) [1] A__ADX(cons(head(z01), z1)) -> c4(A__INCR(cons(a__head(mark(z01)), adx(z1))), MARK(head(z01))) [1] A__ADX(cons(tail(z02), z1)) -> c4(A__INCR(cons(a__tail(mark(z02)), adx(z1))), MARK(tail(z02))) [1] A__ADX(cons(nil, z1)) -> c4(A__INCR(cons(nil, adx(z1))), MARK(nil)) [1] A__ADX(cons(cons(z03, z1'), z1)) -> c4(A__INCR(cons(cons(mark(z03), z1'), adx(z1))), MARK(cons(z03, z1'))) [1] A__ADX(cons(s(z04), z1)) -> c4(A__INCR(cons(s(mark(z04)), adx(z1))), MARK(s(z04))) [1] A__ADX(cons(0, z1)) -> c4(A__INCR(cons(0, adx(z1))), MARK(0)) [1] A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(null_mark, adx(z1))), MARK(z0)) [1] A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) [1] A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) [1] MARK(incr(incr(z05))) -> c14(A__INCR(a__incr(mark(z05))), MARK(incr(z05))) [1] MARK(incr(adx(z06))) -> c14(A__INCR(a__adx(mark(z06))), MARK(adx(z06))) [1] MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) [1] MARK(incr(zeros)) -> c14(A__INCR(a__zeros), MARK(zeros)) [1] MARK(incr(head(z07))) -> c14(A__INCR(a__head(mark(z07))), MARK(head(z07))) [1] MARK(incr(tail(z08))) -> c14(A__INCR(a__tail(mark(z08))), MARK(tail(z08))) [1] MARK(incr(nil)) -> c14(A__INCR(nil), MARK(nil)) [1] MARK(incr(cons(z09, z1''))) -> c14(A__INCR(cons(mark(z09), z1'')), MARK(cons(z09, z1''))) [1] MARK(incr(s(z010))) -> c14(A__INCR(s(mark(z010))), MARK(s(z010))) [1] MARK(incr(0)) -> c14(A__INCR(0), MARK(0)) [1] MARK(incr(z0)) -> c14(A__INCR(null_mark), MARK(z0)) [1] MARK(adx(incr(z011))) -> c15(A__ADX(a__incr(mark(z011))), MARK(incr(z011))) [1] MARK(adx(adx(z012))) -> c15(A__ADX(a__adx(mark(z012))), MARK(adx(z012))) [1] MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) [1] MARK(adx(zeros)) -> c15(A__ADX(a__zeros), MARK(zeros)) [1] MARK(adx(head(z013))) -> c15(A__ADX(a__head(mark(z013))), MARK(head(z013))) [1] MARK(adx(tail(z014))) -> c15(A__ADX(a__tail(mark(z014))), MARK(tail(z014))) [1] MARK(adx(nil)) -> c15(A__ADX(nil), MARK(nil)) [1] MARK(adx(cons(z015, z11))) -> c15(A__ADX(cons(mark(z015), z11)), MARK(cons(z015, z11))) [1] MARK(adx(s(z016))) -> c15(A__ADX(s(mark(z016))), MARK(s(z016))) [1] MARK(adx(0)) -> c15(A__ADX(0), MARK(0)) [1] MARK(adx(z0)) -> c15(A__ADX(null_mark), MARK(z0)) [1] MARK(nats) -> c16(A__NATS) [1] MARK(head(incr(z017))) -> c18(A__HEAD(a__incr(mark(z017))), MARK(incr(z017))) [1] MARK(head(adx(z018))) -> c18(A__HEAD(a__adx(mark(z018))), MARK(adx(z018))) [1] MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) [1] MARK(head(zeros)) -> c18(A__HEAD(a__zeros), MARK(zeros)) [1] MARK(head(head(z019))) -> c18(A__HEAD(a__head(mark(z019))), MARK(head(z019))) [1] MARK(head(tail(z020))) -> c18(A__HEAD(a__tail(mark(z020))), MARK(tail(z020))) [1] MARK(head(nil)) -> c18(A__HEAD(nil), MARK(nil)) [1] MARK(head(cons(z021, z12))) -> c18(A__HEAD(cons(mark(z021), z12)), MARK(cons(z021, z12))) [1] MARK(head(s(z022))) -> c18(A__HEAD(s(mark(z022))), MARK(s(z022))) [1] MARK(head(0)) -> c18(A__HEAD(0), MARK(0)) [1] MARK(head(z0)) -> c18(A__HEAD(null_mark), MARK(z0)) [1] MARK(tail(incr(z023))) -> c19(A__TAIL(a__incr(mark(z023))), MARK(incr(z023))) [1] MARK(tail(adx(z024))) -> c19(A__TAIL(a__adx(mark(z024))), MARK(adx(z024))) [1] MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) [1] MARK(tail(zeros)) -> c19(A__TAIL(a__zeros), MARK(zeros)) [1] MARK(tail(head(z025))) -> c19(A__TAIL(a__head(mark(z025))), MARK(head(z025))) [1] MARK(tail(tail(z026))) -> c19(A__TAIL(a__tail(mark(z026))), MARK(tail(z026))) [1] MARK(tail(nil)) -> c19(A__TAIL(nil), MARK(nil)) [1] MARK(tail(cons(z027, z13))) -> c19(A__TAIL(cons(mark(z027), z13)), MARK(cons(z027, z13))) [1] MARK(tail(s(z028))) -> c19(A__TAIL(s(mark(z028))), MARK(s(z028))) [1] MARK(tail(0)) -> c19(A__TAIL(0), MARK(0)) [1] MARK(tail(z0)) -> c19(A__TAIL(null_mark), MARK(z0)) [1] MARK(cons(z0, z1)) -> c21(MARK(z0)) [1] MARK(s(z0)) -> c22(MARK(z0)) [1] A__NATS -> c6(A__ADX(cons(0, zeros))) [1] A__NATS -> c6(A__ADX(zeros)) [1] A__NATS -> c6(A__ADX(null_a__zeros)) [1] a__incr(nil) -> nil [0] a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) [0] a__incr(z0) -> incr(z0) [0] a__adx(nil) -> nil [0] a__adx(cons(incr(z029), z1)) -> a__incr(cons(a__incr(mark(z029)), adx(z1))) [0] a__adx(cons(adx(z030), z1)) -> a__incr(cons(a__adx(mark(z030)), adx(z1))) [0] a__adx(cons(nats, z1)) -> a__incr(cons(a__nats, adx(z1))) [0] a__adx(cons(zeros, z1)) -> a__incr(cons(a__zeros, adx(z1))) [0] a__adx(cons(head(z031), z1)) -> a__incr(cons(a__head(mark(z031)), adx(z1))) [0] a__adx(cons(tail(z032), z1)) -> a__incr(cons(a__tail(mark(z032)), adx(z1))) [0] a__adx(cons(nil, z1)) -> a__incr(cons(nil, adx(z1))) [0] a__adx(cons(cons(z033, z14), z1)) -> a__incr(cons(cons(mark(z033), z14), adx(z1))) [0] a__adx(cons(s(z034), z1)) -> a__incr(cons(s(mark(z034)), adx(z1))) [0] a__adx(cons(0, z1)) -> a__incr(cons(0, adx(z1))) [0] a__adx(cons(z0, z1)) -> a__incr(cons(null_mark, adx(z1))) [0] a__adx(z0) -> adx(z0) [0] a__nats -> a__adx(cons(0, zeros)) [0] a__nats -> a__adx(zeros) [0] a__nats -> a__adx(null_a__zeros) [0] a__nats -> nats [0] a__zeros -> cons(0, zeros) [0] a__zeros -> zeros [0] a__head(cons(z0, z1)) -> mark(z0) [0] a__head(z0) -> head(z0) [0] a__tail(cons(z0, z1)) -> mark(z1) [0] a__tail(z0) -> tail(z0) [0] mark(incr(incr(z035))) -> a__incr(a__incr(mark(z035))) [0] mark(incr(adx(z036))) -> a__incr(a__adx(mark(z036))) [0] mark(incr(nats)) -> a__incr(a__nats) [0] mark(incr(zeros)) -> a__incr(a__zeros) [0] mark(incr(head(z037))) -> a__incr(a__head(mark(z037))) [0] mark(incr(tail(z038))) -> a__incr(a__tail(mark(z038))) [0] mark(incr(nil)) -> a__incr(nil) [0] mark(incr(cons(z039, z15))) -> a__incr(cons(mark(z039), z15)) [0] mark(incr(s(z040))) -> a__incr(s(mark(z040))) [0] mark(incr(0)) -> a__incr(0) [0] mark(incr(z0)) -> a__incr(null_mark) [0] mark(adx(incr(z041))) -> a__adx(a__incr(mark(z041))) [0] mark(adx(adx(z042))) -> a__adx(a__adx(mark(z042))) [0] mark(adx(nats)) -> a__adx(a__nats) [0] mark(adx(zeros)) -> a__adx(a__zeros) [0] mark(adx(head(z043))) -> a__adx(a__head(mark(z043))) [0] mark(adx(tail(z044))) -> a__adx(a__tail(mark(z044))) [0] mark(adx(nil)) -> a__adx(nil) [0] mark(adx(cons(z045, z16))) -> a__adx(cons(mark(z045), z16)) [0] mark(adx(s(z046))) -> a__adx(s(mark(z046))) [0] mark(adx(0)) -> a__adx(0) [0] mark(adx(z0)) -> a__adx(null_mark) [0] mark(nats) -> a__nats [0] mark(zeros) -> a__zeros [0] mark(head(incr(z047))) -> a__head(a__incr(mark(z047))) [0] mark(head(adx(z048))) -> a__head(a__adx(mark(z048))) [0] mark(head(nats)) -> a__head(a__nats) [0] mark(head(zeros)) -> a__head(a__zeros) [0] mark(head(head(z049))) -> a__head(a__head(mark(z049))) [0] mark(head(tail(z050))) -> a__head(a__tail(mark(z050))) [0] mark(head(nil)) -> a__head(nil) [0] mark(head(cons(z051, z17))) -> a__head(cons(mark(z051), z17)) [0] mark(head(s(z052))) -> a__head(s(mark(z052))) [0] mark(head(0)) -> a__head(0) [0] mark(head(z0)) -> a__head(null_mark) [0] mark(tail(incr(z053))) -> a__tail(a__incr(mark(z053))) [0] mark(tail(adx(z054))) -> a__tail(a__adx(mark(z054))) [0] mark(tail(nats)) -> a__tail(a__nats) [0] mark(tail(zeros)) -> a__tail(a__zeros) [0] mark(tail(head(z055))) -> a__tail(a__head(mark(z055))) [0] mark(tail(tail(z056))) -> a__tail(a__tail(mark(z056))) [0] mark(tail(nil)) -> a__tail(nil) [0] mark(tail(cons(z057, z18))) -> a__tail(cons(mark(z057), z18)) [0] mark(tail(s(z058))) -> a__tail(s(mark(z058))) [0] mark(tail(0)) -> a__tail(0) [0] mark(tail(z0)) -> a__tail(null_mark) [0] mark(nil) -> nil [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(s(z0)) -> s(mark(z0)) [0] mark(0) -> 0 [0] a__incr(v0) -> null_a__incr [0] a__adx(v0) -> null_a__adx [0] a__nats -> null_a__nats [0] a__zeros -> null_a__zeros [0] a__head(v0) -> null_a__head [0] a__tail(v0) -> null_a__tail [0] mark(v0) -> null_mark [0] The TRS has the following type information: A__INCR :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> c1 cons :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c1 :: c14:c15:c16:c18:c19:c21:c22 -> c1 MARK :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> c14:c15:c16:c18:c19:c21:c22 A__ADX :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> c4 c4 :: c1 -> c14:c15:c16:c18:c19:c21:c22 -> c4 mark :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark adx :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark A__HEAD :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> c10 c10 :: c14:c15:c16:c18:c19:c21:c22 -> c10 A__TAIL :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> c12 c12 :: c14:c15:c16:c18:c19:c21:c22 -> c12 incr :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c14 :: c1 -> c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 c15 :: c4 -> c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 nats :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c16 :: c6 -> c14:c15:c16:c18:c19:c21:c22 A__NATS :: c6 head :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c18 :: c10 -> c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 tail :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c19 :: c12 -> c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 c21 :: c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 s :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c22 :: c14:c15:c16:c18:c19:c21:c22 -> c14:c15:c16:c18:c19:c21:c22 c6 :: c4 -> c6 a__zeros :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark a__incr :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark nil :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark a__adx :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark a__nats :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark 0 :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark zeros :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark a__head :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark a__tail :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__incr :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__adx :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__nats :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__zeros :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__head :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__tail :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_mark :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark const :: c1 const1 :: c14:c15:c16:c18:c19:c21:c22 const2 :: c4 const3 :: c10 const4 :: c12 const5 :: c6 Rewrite Strategy: INNERMOST ---------------------------------------- (41) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nats => 1 nil => 2 0 => 0 zeros => 3 null_a__incr => 0 null_a__adx => 0 null_a__nats => 0 null_a__zeros => 0 null_a__head => 0 null_a__tail => 0 null_mark => 0 const => 0 const1 => 0 const2 => 0 const3 => 0 const4 => 0 const5 => 0 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__zeros + (1 + z1)) + MARK(3) :|: z1 >= 0, z = 1 + 3 + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__tail(mark(z02)) + (1 + z1)) + MARK(1 + z02) :|: z1 >= 0, z = 1 + (1 + z02) + z1, z02 >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__nats + (1 + z1)) + MARK(1) :|: z1 >= 0, z = 1 + 1 + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__incr(mark(z0')) + (1 + z1)) + MARK(1 + z0') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0') + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__head(mark(z01)) + (1 + z1)) + MARK(1 + z01) :|: z = 1 + (1 + z01) + z1, z1 >= 0, z01 >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__adx(mark(z0'')) + (1 + z1)) + MARK(1 + z0'') :|: z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + 2 + (1 + z1)) + MARK(2) :|: z1 >= 0, z = 1 + 2 + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + 0 + (1 + z1)) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + 0 + (1 + z1)) + MARK(0) :|: z1 >= 0, z = 1 + 0 + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + (1 + mark(z04)) + (1 + z1)) + MARK(1 + z04) :|: z04 >= 0, z1 >= 0, z = 1 + (1 + z04) + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + (1 + mark(z03) + z1') + (1 + z1)) + MARK(1 + z03 + z1') :|: z1 >= 0, z1' >= 0, z = 1 + (1 + z03 + z1') + z1, z03 >= 0 A__HEAD(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__INCR(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__NATS -{ 1 }-> 1 + A__ADX(3) :|: A__NATS -{ 1 }-> 1 + A__ADX(0) :|: A__NATS -{ 1 }-> 1 + A__ADX(1 + 0 + 3) :|: A__TAIL(z) -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__NATS :|: z = 1 MARK(z) -{ 1 }-> 1 + A__TAIL(a__zeros) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__TAIL(a__tail(mark(z026))) + MARK(1 + z026) :|: z = 1 + (1 + z026), z026 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(a__nats) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__TAIL(a__incr(mark(z023))) + MARK(1 + z023) :|: z023 >= 0, z = 1 + (1 + z023) MARK(z) -{ 1 }-> 1 + A__TAIL(a__head(mark(z025))) + MARK(1 + z025) :|: z025 >= 0, z = 1 + (1 + z025) MARK(z) -{ 1 }-> 1 + A__TAIL(a__adx(mark(z024))) + MARK(1 + z024) :|: z024 >= 0, z = 1 + (1 + z024) MARK(z) -{ 1 }-> 1 + A__TAIL(2) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__TAIL(0) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__TAIL(1 + mark(z028)) + MARK(1 + z028) :|: z028 >= 0, z = 1 + (1 + z028) MARK(z) -{ 1 }-> 1 + A__TAIL(1 + mark(z027) + z13) + MARK(1 + z027 + z13) :|: z027 >= 0, z = 1 + (1 + z027 + z13), z13 >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(a__zeros) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__INCR(a__tail(mark(z08))) + MARK(1 + z08) :|: z08 >= 0, z = 1 + (1 + z08) MARK(z) -{ 1 }-> 1 + A__INCR(a__nats) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__INCR(a__incr(mark(z05))) + MARK(1 + z05) :|: z05 >= 0, z = 1 + (1 + z05) MARK(z) -{ 1 }-> 1 + A__INCR(a__head(mark(z07))) + MARK(1 + z07) :|: z = 1 + (1 + z07), z07 >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(a__adx(mark(z06))) + MARK(1 + z06) :|: z06 >= 0, z = 1 + (1 + z06) MARK(z) -{ 1 }-> 1 + A__INCR(2) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__INCR(0) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__INCR(1 + mark(z010)) + MARK(1 + z010) :|: z = 1 + (1 + z010), z010 >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(1 + mark(z09) + z1'') + MARK(1 + z09 + z1'') :|: z = 1 + (1 + z09 + z1''), z09 >= 0, z1'' >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__zeros) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__HEAD(a__tail(mark(z020))) + MARK(1 + z020) :|: z = 1 + (1 + z020), z020 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__nats) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__HEAD(a__incr(mark(z017))) + MARK(1 + z017) :|: z = 1 + (1 + z017), z017 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__head(mark(z019))) + MARK(1 + z019) :|: z019 >= 0, z = 1 + (1 + z019) MARK(z) -{ 1 }-> 1 + A__HEAD(a__adx(mark(z018))) + MARK(1 + z018) :|: z = 1 + (1 + z018), z018 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(2) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__HEAD(0) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__HEAD(1 + mark(z022)) + MARK(1 + z022) :|: z022 >= 0, z = 1 + (1 + z022) MARK(z) -{ 1 }-> 1 + A__HEAD(1 + mark(z021) + z12) + MARK(1 + z021 + z12) :|: z021 >= 0, z = 1 + (1 + z021 + z12), z12 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(a__zeros) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__ADX(a__tail(mark(z014))) + MARK(1 + z014) :|: z = 1 + (1 + z014), z014 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(a__nats) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__ADX(a__incr(mark(z011))) + MARK(1 + z011) :|: z = 1 + (1 + z011), z011 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(a__head(mark(z013))) + MARK(1 + z013) :|: z013 >= 0, z = 1 + (1 + z013) MARK(z) -{ 1 }-> 1 + A__ADX(a__adx(mark(z012))) + MARK(1 + z012) :|: z012 >= 0, z = 1 + (1 + z012) MARK(z) -{ 1 }-> 1 + A__ADX(2) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__ADX(0) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__ADX(1 + mark(z016)) + MARK(1 + z016) :|: z016 >= 0, z = 1 + (1 + z016) MARK(z) -{ 1 }-> 1 + A__ADX(1 + mark(z015) + z11) + MARK(1 + z015 + z11) :|: z = 1 + (1 + z015 + z11), z015 >= 0, z11 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + a__zeros + (1 + z1)) :|: z1 >= 0, z = 1 + 3 + z1 a__adx(z) -{ 0 }-> a__incr(1 + a__tail(mark(z032)) + (1 + z1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1 a__adx(z) -{ 0 }-> a__incr(1 + a__nats + (1 + z1)) :|: z1 >= 0, z = 1 + 1 + z1 a__adx(z) -{ 0 }-> a__incr(1 + a__incr(mark(z029)) + (1 + z1)) :|: z029 >= 0, z = 1 + (1 + z029) + z1, z1 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + a__head(mark(z031)) + (1 + z1)) :|: z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + a__adx(mark(z030)) + (1 + z1)) :|: z1 >= 0, z = 1 + (1 + z030) + z1, z030 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + 2 + (1 + z1)) :|: z1 >= 0, z = 1 + 2 + z1 a__adx(z) -{ 0 }-> a__incr(1 + 0 + (1 + z1)) :|: z1 >= 0, z = 1 + 0 + z1 a__adx(z) -{ 0 }-> a__incr(1 + 0 + (1 + z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__adx(z) -{ 0 }-> a__incr(1 + (1 + mark(z034)) + (1 + z1)) :|: z1 >= 0, z034 >= 0, z = 1 + (1 + z034) + z1 a__adx(z) -{ 0 }-> a__incr(1 + (1 + mark(z033) + z14) + (1 + z1)) :|: z1 >= 0, z033 >= 0, z = 1 + (1 + z033 + z14) + z1, z14 >= 0 a__adx(z) -{ 0 }-> 2 :|: z = 2 a__adx(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__adx(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__head(z) -{ 0 }-> mark(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__head(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__incr(z) -{ 0 }-> 2 :|: z = 2 a__incr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__incr(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__incr(z) -{ 0 }-> 1 + (1 + mark(z0)) + (1 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__nats -{ 0 }-> a__adx(3) :|: a__nats -{ 0 }-> a__adx(0) :|: a__nats -{ 0 }-> a__adx(1 + 0 + 3) :|: a__nats -{ 0 }-> 1 :|: a__nats -{ 0 }-> 0 :|: a__tail(z) -{ 0 }-> mark(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__tail(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__zeros -{ 0 }-> 3 :|: a__zeros -{ 0 }-> 0 :|: a__zeros -{ 0 }-> 1 + 0 + 3 :|: mark(z) -{ 0 }-> a__zeros :|: z = 3 mark(z) -{ 0 }-> a__tail(a__zeros) :|: z = 1 + 3 mark(z) -{ 0 }-> a__tail(a__tail(mark(z056))) :|: z056 >= 0, z = 1 + (1 + z056) mark(z) -{ 0 }-> a__tail(a__nats) :|: z = 1 + 1 mark(z) -{ 0 }-> a__tail(a__incr(mark(z053))) :|: z053 >= 0, z = 1 + (1 + z053) mark(z) -{ 0 }-> a__tail(a__head(mark(z055))) :|: z = 1 + (1 + z055), z055 >= 0 mark(z) -{ 0 }-> a__tail(a__adx(mark(z054))) :|: z054 >= 0, z = 1 + (1 + z054) mark(z) -{ 0 }-> a__tail(2) :|: z = 1 + 2 mark(z) -{ 0 }-> a__tail(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__tail(0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__tail(1 + mark(z058)) :|: z = 1 + (1 + z058), z058 >= 0 mark(z) -{ 0 }-> a__tail(1 + mark(z057) + z18) :|: z18 >= 0, z = 1 + (1 + z057 + z18), z057 >= 0 mark(z) -{ 0 }-> a__nats :|: z = 1 mark(z) -{ 0 }-> a__incr(a__zeros) :|: z = 1 + 3 mark(z) -{ 0 }-> a__incr(a__tail(mark(z038))) :|: z = 1 + (1 + z038), z038 >= 0 mark(z) -{ 0 }-> a__incr(a__nats) :|: z = 1 + 1 mark(z) -{ 0 }-> a__incr(a__incr(mark(z035))) :|: z035 >= 0, z = 1 + (1 + z035) mark(z) -{ 0 }-> a__incr(a__head(mark(z037))) :|: z = 1 + (1 + z037), z037 >= 0 mark(z) -{ 0 }-> a__incr(a__adx(mark(z036))) :|: z036 >= 0, z = 1 + (1 + z036) mark(z) -{ 0 }-> a__incr(2) :|: z = 1 + 2 mark(z) -{ 0 }-> a__incr(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__incr(0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__incr(1 + mark(z040)) :|: z = 1 + (1 + z040), z040 >= 0 mark(z) -{ 0 }-> a__incr(1 + mark(z039) + z15) :|: z15 >= 0, z039 >= 0, z = 1 + (1 + z039 + z15) mark(z) -{ 0 }-> a__head(a__zeros) :|: z = 1 + 3 mark(z) -{ 0 }-> a__head(a__tail(mark(z050))) :|: z = 1 + (1 + z050), z050 >= 0 mark(z) -{ 0 }-> a__head(a__nats) :|: z = 1 + 1 mark(z) -{ 0 }-> a__head(a__incr(mark(z047))) :|: z = 1 + (1 + z047), z047 >= 0 mark(z) -{ 0 }-> a__head(a__head(mark(z049))) :|: z = 1 + (1 + z049), z049 >= 0 mark(z) -{ 0 }-> a__head(a__adx(mark(z048))) :|: z048 >= 0, z = 1 + (1 + z048) mark(z) -{ 0 }-> a__head(2) :|: z = 1 + 2 mark(z) -{ 0 }-> a__head(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__head(0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__head(1 + mark(z052)) :|: z052 >= 0, z = 1 + (1 + z052) mark(z) -{ 0 }-> a__head(1 + mark(z051) + z17) :|: z17 >= 0, z051 >= 0, z = 1 + (1 + z051 + z17) mark(z) -{ 0 }-> a__adx(a__zeros) :|: z = 1 + 3 mark(z) -{ 0 }-> a__adx(a__tail(mark(z044))) :|: z044 >= 0, z = 1 + (1 + z044) mark(z) -{ 0 }-> a__adx(a__nats) :|: z = 1 + 1 mark(z) -{ 0 }-> a__adx(a__incr(mark(z041))) :|: z = 1 + (1 + z041), z041 >= 0 mark(z) -{ 0 }-> a__adx(a__head(mark(z043))) :|: z043 >= 0, z = 1 + (1 + z043) mark(z) -{ 0 }-> a__adx(a__adx(mark(z042))) :|: z042 >= 0, z = 1 + (1 + z042) mark(z) -{ 0 }-> a__adx(2) :|: z = 1 + 2 mark(z) -{ 0 }-> a__adx(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__adx(0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__adx(1 + mark(z046)) :|: z046 >= 0, z = 1 + (1 + z046) mark(z) -{ 0 }-> a__adx(1 + mark(z045) + z16) :|: z = 1 + (1 + z045 + z16), z045 >= 0, z16 >= 0 mark(z) -{ 0 }-> 2 :|: z = 2 mark(z) -{ 0 }-> 0 :|: z = 0 mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 mark(z) -{ 0 }-> 1 + mark(z0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> 1 + mark(z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 ---------------------------------------- (43) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: a__zeros -{ 0 }-> 3 :|: a__zeros -{ 0 }-> 0 :|: a__zeros -{ 0 }-> 1 + 0 + 3 :|: ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__tail(mark(z02)) + (1 + z1)) + MARK(1 + z02) :|: z1 >= 0, z = 1 + (1 + z02) + z1, z02 >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__nats + (1 + z1)) + MARK(1) :|: z1 >= 0, z = 1 + 1 + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__incr(mark(z0')) + (1 + z1)) + MARK(1 + z0') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0') + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__head(mark(z01)) + (1 + z1)) + MARK(1 + z01) :|: z = 1 + (1 + z01) + z1, z1 >= 0, z01 >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__adx(mark(z0'')) + (1 + z1)) + MARK(1 + z0'') :|: z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + 3 + (1 + z1)) + MARK(3) :|: z1 >= 0, z = 1 + 3 + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + 2 + (1 + z1)) + MARK(2) :|: z1 >= 0, z = 1 + 2 + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + 0 + (1 + z1)) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + 0 + (1 + z1)) + MARK(3) :|: z1 >= 0, z = 1 + 3 + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + 0 + (1 + z1)) + MARK(0) :|: z1 >= 0, z = 1 + 0 + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + (1 + mark(z04)) + (1 + z1)) + MARK(1 + z04) :|: z04 >= 0, z1 >= 0, z = 1 + (1 + z04) + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + (1 + mark(z03) + z1') + (1 + z1)) + MARK(1 + z03 + z1') :|: z1 >= 0, z1' >= 0, z = 1 + (1 + z03 + z1') + z1, z03 >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + (1 + 0 + 3) + (1 + z1)) + MARK(3) :|: z1 >= 0, z = 1 + 3 + z1 A__HEAD(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__INCR(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__NATS -{ 1 }-> 1 + A__ADX(3) :|: A__NATS -{ 1 }-> 1 + A__ADX(0) :|: A__NATS -{ 1 }-> 1 + A__ADX(1 + 0 + 3) :|: A__TAIL(z) -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__NATS :|: z = 1 MARK(z) -{ 1 }-> 1 + A__TAIL(a__tail(mark(z026))) + MARK(1 + z026) :|: z = 1 + (1 + z026), z026 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(a__nats) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__TAIL(a__incr(mark(z023))) + MARK(1 + z023) :|: z023 >= 0, z = 1 + (1 + z023) MARK(z) -{ 1 }-> 1 + A__TAIL(a__head(mark(z025))) + MARK(1 + z025) :|: z025 >= 0, z = 1 + (1 + z025) MARK(z) -{ 1 }-> 1 + A__TAIL(a__adx(mark(z024))) + MARK(1 + z024) :|: z024 >= 0, z = 1 + (1 + z024) MARK(z) -{ 1 }-> 1 + A__TAIL(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__TAIL(2) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__TAIL(0) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(0) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__TAIL(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__TAIL(1 + mark(z028)) + MARK(1 + z028) :|: z028 >= 0, z = 1 + (1 + z028) MARK(z) -{ 1 }-> 1 + A__TAIL(1 + mark(z027) + z13) + MARK(1 + z027 + z13) :|: z027 >= 0, z = 1 + (1 + z027 + z13), z13 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(1 + 0 + 3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__INCR(a__tail(mark(z08))) + MARK(1 + z08) :|: z08 >= 0, z = 1 + (1 + z08) MARK(z) -{ 1 }-> 1 + A__INCR(a__nats) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__INCR(a__incr(mark(z05))) + MARK(1 + z05) :|: z05 >= 0, z = 1 + (1 + z05) MARK(z) -{ 1 }-> 1 + A__INCR(a__head(mark(z07))) + MARK(1 + z07) :|: z = 1 + (1 + z07), z07 >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(a__adx(mark(z06))) + MARK(1 + z06) :|: z06 >= 0, z = 1 + (1 + z06) MARK(z) -{ 1 }-> 1 + A__INCR(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__INCR(2) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__INCR(0) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(0) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__INCR(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__INCR(1 + mark(z010)) + MARK(1 + z010) :|: z = 1 + (1 + z010), z010 >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(1 + mark(z09) + z1'') + MARK(1 + z09 + z1'') :|: z = 1 + (1 + z09 + z1''), z09 >= 0, z1'' >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(1 + 0 + 3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__HEAD(a__tail(mark(z020))) + MARK(1 + z020) :|: z = 1 + (1 + z020), z020 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__nats) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__HEAD(a__incr(mark(z017))) + MARK(1 + z017) :|: z = 1 + (1 + z017), z017 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__head(mark(z019))) + MARK(1 + z019) :|: z019 >= 0, z = 1 + (1 + z019) MARK(z) -{ 1 }-> 1 + A__HEAD(a__adx(mark(z018))) + MARK(1 + z018) :|: z = 1 + (1 + z018), z018 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__HEAD(2) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__HEAD(0) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(0) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__HEAD(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__HEAD(1 + mark(z022)) + MARK(1 + z022) :|: z022 >= 0, z = 1 + (1 + z022) MARK(z) -{ 1 }-> 1 + A__HEAD(1 + mark(z021) + z12) + MARK(1 + z021 + z12) :|: z021 >= 0, z = 1 + (1 + z021 + z12), z12 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(1 + 0 + 3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__ADX(a__tail(mark(z014))) + MARK(1 + z014) :|: z = 1 + (1 + z014), z014 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(a__nats) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__ADX(a__incr(mark(z011))) + MARK(1 + z011) :|: z = 1 + (1 + z011), z011 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(a__head(mark(z013))) + MARK(1 + z013) :|: z013 >= 0, z = 1 + (1 + z013) MARK(z) -{ 1 }-> 1 + A__ADX(a__adx(mark(z012))) + MARK(1 + z012) :|: z012 >= 0, z = 1 + (1 + z012) MARK(z) -{ 1 }-> 1 + A__ADX(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__ADX(2) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__ADX(0) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(0) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__ADX(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__ADX(1 + mark(z016)) + MARK(1 + z016) :|: z016 >= 0, z = 1 + (1 + z016) MARK(z) -{ 1 }-> 1 + A__ADX(1 + mark(z015) + z11) + MARK(1 + z015 + z11) :|: z = 1 + (1 + z015 + z11), z015 >= 0, z11 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(1 + 0 + 3) + MARK(3) :|: z = 1 + 3 a__adx(z) -{ 0 }-> a__incr(1 + a__tail(mark(z032)) + (1 + z1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1 a__adx(z) -{ 0 }-> a__incr(1 + a__nats + (1 + z1)) :|: z1 >= 0, z = 1 + 1 + z1 a__adx(z) -{ 0 }-> a__incr(1 + a__incr(mark(z029)) + (1 + z1)) :|: z029 >= 0, z = 1 + (1 + z029) + z1, z1 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + a__head(mark(z031)) + (1 + z1)) :|: z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + a__adx(mark(z030)) + (1 + z1)) :|: z1 >= 0, z = 1 + (1 + z030) + z1, z030 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + 3 + (1 + z1)) :|: z1 >= 0, z = 1 + 3 + z1 a__adx(z) -{ 0 }-> a__incr(1 + 2 + (1 + z1)) :|: z1 >= 0, z = 1 + 2 + z1 a__adx(z) -{ 0 }-> a__incr(1 + 0 + (1 + z1)) :|: z1 >= 0, z = 1 + 0 + z1 a__adx(z) -{ 0 }-> a__incr(1 + 0 + (1 + z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__adx(z) -{ 0 }-> a__incr(1 + 0 + (1 + z1)) :|: z1 >= 0, z = 1 + 3 + z1 a__adx(z) -{ 0 }-> a__incr(1 + (1 + mark(z034)) + (1 + z1)) :|: z1 >= 0, z034 >= 0, z = 1 + (1 + z034) + z1 a__adx(z) -{ 0 }-> a__incr(1 + (1 + mark(z033) + z14) + (1 + z1)) :|: z1 >= 0, z033 >= 0, z = 1 + (1 + z033 + z14) + z1, z14 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + (1 + 0 + 3) + (1 + z1)) :|: z1 >= 0, z = 1 + 3 + z1 a__adx(z) -{ 0 }-> 2 :|: z = 2 a__adx(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__adx(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__head(z) -{ 0 }-> mark(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__head(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__incr(z) -{ 0 }-> 2 :|: z = 2 a__incr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__incr(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__incr(z) -{ 0 }-> 1 + (1 + mark(z0)) + (1 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__nats -{ 0 }-> a__adx(3) :|: a__nats -{ 0 }-> a__adx(0) :|: a__nats -{ 0 }-> a__adx(1 + 0 + 3) :|: a__nats -{ 0 }-> 1 :|: a__nats -{ 0 }-> 0 :|: a__tail(z) -{ 0 }-> mark(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__tail(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__zeros -{ 0 }-> 3 :|: a__zeros -{ 0 }-> 0 :|: a__zeros -{ 0 }-> 1 + 0 + 3 :|: mark(z) -{ 0 }-> a__tail(a__tail(mark(z056))) :|: z056 >= 0, z = 1 + (1 + z056) mark(z) -{ 0 }-> a__tail(a__nats) :|: z = 1 + 1 mark(z) -{ 0 }-> a__tail(a__incr(mark(z053))) :|: z053 >= 0, z = 1 + (1 + z053) mark(z) -{ 0 }-> a__tail(a__head(mark(z055))) :|: z = 1 + (1 + z055), z055 >= 0 mark(z) -{ 0 }-> a__tail(a__adx(mark(z054))) :|: z054 >= 0, z = 1 + (1 + z054) mark(z) -{ 0 }-> a__tail(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__tail(2) :|: z = 1 + 2 mark(z) -{ 0 }-> a__tail(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__tail(0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__tail(0) :|: z = 1 + 3 mark(z) -{ 0 }-> a__tail(1 + mark(z058)) :|: z = 1 + (1 + z058), z058 >= 0 mark(z) -{ 0 }-> a__tail(1 + mark(z057) + z18) :|: z18 >= 0, z = 1 + (1 + z057 + z18), z057 >= 0 mark(z) -{ 0 }-> a__tail(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__nats :|: z = 1 mark(z) -{ 0 }-> a__incr(a__tail(mark(z038))) :|: z = 1 + (1 + z038), z038 >= 0 mark(z) -{ 0 }-> a__incr(a__nats) :|: z = 1 + 1 mark(z) -{ 0 }-> a__incr(a__incr(mark(z035))) :|: z035 >= 0, z = 1 + (1 + z035) mark(z) -{ 0 }-> a__incr(a__head(mark(z037))) :|: z = 1 + (1 + z037), z037 >= 0 mark(z) -{ 0 }-> a__incr(a__adx(mark(z036))) :|: z036 >= 0, z = 1 + (1 + z036) mark(z) -{ 0 }-> a__incr(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__incr(2) :|: z = 1 + 2 mark(z) -{ 0 }-> a__incr(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__incr(0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__incr(0) :|: z = 1 + 3 mark(z) -{ 0 }-> a__incr(1 + mark(z040)) :|: z = 1 + (1 + z040), z040 >= 0 mark(z) -{ 0 }-> a__incr(1 + mark(z039) + z15) :|: z15 >= 0, z039 >= 0, z = 1 + (1 + z039 + z15) mark(z) -{ 0 }-> a__incr(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__head(a__tail(mark(z050))) :|: z = 1 + (1 + z050), z050 >= 0 mark(z) -{ 0 }-> a__head(a__nats) :|: z = 1 + 1 mark(z) -{ 0 }-> a__head(a__incr(mark(z047))) :|: z = 1 + (1 + z047), z047 >= 0 mark(z) -{ 0 }-> a__head(a__head(mark(z049))) :|: z = 1 + (1 + z049), z049 >= 0 mark(z) -{ 0 }-> a__head(a__adx(mark(z048))) :|: z048 >= 0, z = 1 + (1 + z048) mark(z) -{ 0 }-> a__head(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__head(2) :|: z = 1 + 2 mark(z) -{ 0 }-> a__head(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__head(0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__head(0) :|: z = 1 + 3 mark(z) -{ 0 }-> a__head(1 + mark(z052)) :|: z052 >= 0, z = 1 + (1 + z052) mark(z) -{ 0 }-> a__head(1 + mark(z051) + z17) :|: z17 >= 0, z051 >= 0, z = 1 + (1 + z051 + z17) mark(z) -{ 0 }-> a__head(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__adx(a__tail(mark(z044))) :|: z044 >= 0, z = 1 + (1 + z044) mark(z) -{ 0 }-> a__adx(a__nats) :|: z = 1 + 1 mark(z) -{ 0 }-> a__adx(a__incr(mark(z041))) :|: z = 1 + (1 + z041), z041 >= 0 mark(z) -{ 0 }-> a__adx(a__head(mark(z043))) :|: z043 >= 0, z = 1 + (1 + z043) mark(z) -{ 0 }-> a__adx(a__adx(mark(z042))) :|: z042 >= 0, z = 1 + (1 + z042) mark(z) -{ 0 }-> a__adx(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__adx(2) :|: z = 1 + 2 mark(z) -{ 0 }-> a__adx(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__adx(0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__adx(0) :|: z = 1 + 3 mark(z) -{ 0 }-> a__adx(1 + mark(z046)) :|: z046 >= 0, z = 1 + (1 + z046) mark(z) -{ 0 }-> a__adx(1 + mark(z045) + z16) :|: z = 1 + (1 + z045 + z16), z045 >= 0, z16 >= 0 mark(z) -{ 0 }-> a__adx(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 0 }-> 3 :|: z = 3 mark(z) -{ 0 }-> 2 :|: z = 2 mark(z) -{ 0 }-> 0 :|: z = 0 mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 mark(z) -{ 0 }-> 0 :|: z = 3 mark(z) -{ 0 }-> 1 + mark(z0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> 1 + mark(z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> 1 + 0 + 3 :|: z = 3 ---------------------------------------- (45) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__tail(mark(z02)) + (1 + z1)) + MARK(1 + z02) :|: z1 >= 0, z = 1 + (1 + z02) + z1, z02 >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__nats + (1 + (z - 2))) + MARK(1) :|: z - 2 >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__incr(mark(z0')) + (1 + z1)) + MARK(1 + z0') :|: z1 >= 0, z0' >= 0, z = 1 + (1 + z0') + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__head(mark(z01)) + (1 + z1)) + MARK(1 + z01) :|: z = 1 + (1 + z01) + z1, z1 >= 0, z01 >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + a__adx(mark(z0'')) + (1 + z1)) + MARK(1 + z0'') :|: z1 >= 0, z = 1 + (1 + z0'') + z1, z0'' >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + 3 + (1 + (z - 4))) + MARK(3) :|: z - 4 >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + 2 + (1 + (z - 3))) + MARK(2) :|: z - 3 >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + 0 + (1 + z1)) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + 0 + (1 + (z - 4))) + MARK(3) :|: z - 4 >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + 0 + (1 + (z - 1))) + MARK(0) :|: z - 1 >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + (1 + mark(z04)) + (1 + z1)) + MARK(1 + z04) :|: z04 >= 0, z1 >= 0, z = 1 + (1 + z04) + z1 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + (1 + mark(z03) + z1') + (1 + z1)) + MARK(1 + z03 + z1') :|: z1 >= 0, z1' >= 0, z = 1 + (1 + z03 + z1') + z1, z03 >= 0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + (1 + 0 + 3) + (1 + (z - 4))) + MARK(3) :|: z - 4 >= 0 A__HEAD(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__INCR(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__NATS -{ 1 }-> 1 + A__ADX(3) :|: A__NATS -{ 1 }-> 1 + A__ADX(0) :|: A__NATS -{ 1 }-> 1 + A__ADX(1 + 0 + 3) :|: A__TAIL(z) -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z - 1) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__NATS :|: z = 1 MARK(z) -{ 1 }-> 1 + A__TAIL(a__tail(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(a__nats) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__TAIL(a__incr(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(a__head(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(a__adx(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__TAIL(2) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__TAIL(0) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__TAIL(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__TAIL(0) + MARK(z - 1) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(1 + mark(z - 2)) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(1 + mark(z027) + z13) + MARK(1 + z027 + z13) :|: z027 >= 0, z = 1 + (1 + z027 + z13), z13 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(1 + 0 + 3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__INCR(a__tail(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(a__nats) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__INCR(a__incr(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(a__head(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(a__adx(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__INCR(2) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__INCR(0) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__INCR(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__INCR(0) + MARK(z - 1) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(1 + mark(z - 2)) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(1 + mark(z09) + z1'') + MARK(1 + z09 + z1'') :|: z = 1 + (1 + z09 + z1''), z09 >= 0, z1'' >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(1 + 0 + 3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__HEAD(a__tail(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__nats) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__HEAD(a__incr(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__head(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__adx(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__HEAD(2) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__HEAD(0) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__HEAD(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__HEAD(0) + MARK(z - 1) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(1 + mark(z - 2)) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(1 + mark(z021) + z12) + MARK(1 + z021 + z12) :|: z021 >= 0, z = 1 + (1 + z021 + z12), z12 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(1 + 0 + 3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__ADX(a__tail(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(a__nats) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__ADX(a__incr(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(a__head(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(a__adx(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__ADX(2) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__ADX(0) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__ADX(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__ADX(0) + MARK(z - 1) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(1 + mark(z - 2)) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(1 + mark(z015) + z11) + MARK(1 + z015 + z11) :|: z = 1 + (1 + z015 + z11), z015 >= 0, z11 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(1 + 0 + 3) + MARK(3) :|: z = 1 + 3 a__adx(z) -{ 0 }-> a__incr(1 + a__tail(mark(z032)) + (1 + z1)) :|: z032 >= 0, z1 >= 0, z = 1 + (1 + z032) + z1 a__adx(z) -{ 0 }-> a__incr(1 + a__nats + (1 + (z - 2))) :|: z - 2 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + a__incr(mark(z029)) + (1 + z1)) :|: z029 >= 0, z = 1 + (1 + z029) + z1, z1 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + a__head(mark(z031)) + (1 + z1)) :|: z = 1 + (1 + z031) + z1, z031 >= 0, z1 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + a__adx(mark(z030)) + (1 + z1)) :|: z1 >= 0, z = 1 + (1 + z030) + z1, z030 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + 3 + (1 + (z - 4))) :|: z - 4 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + 2 + (1 + (z - 3))) :|: z - 3 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + 0 + (1 + z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__adx(z) -{ 0 }-> a__incr(1 + 0 + (1 + (z - 4))) :|: z - 4 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + 0 + (1 + (z - 1))) :|: z - 1 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + (1 + mark(z034)) + (1 + z1)) :|: z1 >= 0, z034 >= 0, z = 1 + (1 + z034) + z1 a__adx(z) -{ 0 }-> a__incr(1 + (1 + mark(z033) + z14) + (1 + z1)) :|: z1 >= 0, z033 >= 0, z = 1 + (1 + z033 + z14) + z1, z14 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + (1 + 0 + 3) + (1 + (z - 4))) :|: z - 4 >= 0 a__adx(z) -{ 0 }-> 2 :|: z = 2 a__adx(z) -{ 0 }-> 0 :|: z >= 0 a__adx(z) -{ 0 }-> 1 + z :|: z >= 0 a__head(z) -{ 0 }-> mark(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__head(z) -{ 0 }-> 0 :|: z >= 0 a__head(z) -{ 0 }-> 1 + z :|: z >= 0 a__incr(z) -{ 0 }-> 2 :|: z = 2 a__incr(z) -{ 0 }-> 0 :|: z >= 0 a__incr(z) -{ 0 }-> 1 + z :|: z >= 0 a__incr(z) -{ 0 }-> 1 + (1 + mark(z0)) + (1 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__nats -{ 0 }-> a__adx(3) :|: a__nats -{ 0 }-> a__adx(0) :|: a__nats -{ 0 }-> a__adx(1 + 0 + 3) :|: a__nats -{ 0 }-> 1 :|: a__nats -{ 0 }-> 0 :|: a__tail(z) -{ 0 }-> mark(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__tail(z) -{ 0 }-> 0 :|: z >= 0 a__tail(z) -{ 0 }-> 1 + z :|: z >= 0 a__zeros -{ 0 }-> 3 :|: a__zeros -{ 0 }-> 0 :|: a__zeros -{ 0 }-> 1 + 0 + 3 :|: mark(z) -{ 0 }-> a__tail(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__tail(a__nats) :|: z = 1 + 1 mark(z) -{ 0 }-> a__tail(a__incr(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__tail(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__tail(a__adx(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__tail(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__tail(2) :|: z = 1 + 2 mark(z) -{ 0 }-> a__tail(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__tail(0) :|: z - 1 >= 0 mark(z) -{ 0 }-> a__tail(0) :|: z = 1 + 3 mark(z) -{ 0 }-> a__tail(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__tail(1 + mark(z057) + z18) :|: z18 >= 0, z = 1 + (1 + z057 + z18), z057 >= 0 mark(z) -{ 0 }-> a__tail(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__nats :|: z = 1 mark(z) -{ 0 }-> a__incr(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__incr(a__nats) :|: z = 1 + 1 mark(z) -{ 0 }-> a__incr(a__incr(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__incr(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__incr(a__adx(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__incr(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__incr(2) :|: z = 1 + 2 mark(z) -{ 0 }-> a__incr(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__incr(0) :|: z - 1 >= 0 mark(z) -{ 0 }-> a__incr(0) :|: z = 1 + 3 mark(z) -{ 0 }-> a__incr(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__incr(1 + mark(z039) + z15) :|: z15 >= 0, z039 >= 0, z = 1 + (1 + z039 + z15) mark(z) -{ 0 }-> a__incr(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__head(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__head(a__nats) :|: z = 1 + 1 mark(z) -{ 0 }-> a__head(a__incr(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__head(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__head(a__adx(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__head(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__head(2) :|: z = 1 + 2 mark(z) -{ 0 }-> a__head(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__head(0) :|: z - 1 >= 0 mark(z) -{ 0 }-> a__head(0) :|: z = 1 + 3 mark(z) -{ 0 }-> a__head(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__head(1 + mark(z051) + z17) :|: z17 >= 0, z051 >= 0, z = 1 + (1 + z051 + z17) mark(z) -{ 0 }-> a__head(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__adx(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__adx(a__nats) :|: z = 1 + 1 mark(z) -{ 0 }-> a__adx(a__incr(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__adx(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__adx(a__adx(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__adx(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__adx(2) :|: z = 1 + 2 mark(z) -{ 0 }-> a__adx(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__adx(0) :|: z - 1 >= 0 mark(z) -{ 0 }-> a__adx(0) :|: z = 1 + 3 mark(z) -{ 0 }-> a__adx(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__adx(1 + mark(z045) + z16) :|: z = 1 + (1 + z045 + z16), z045 >= 0, z16 >= 0 mark(z) -{ 0 }-> a__adx(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 0 }-> 3 :|: z = 3 mark(z) -{ 0 }-> 2 :|: z = 2 mark(z) -{ 0 }-> 0 :|: z = 0 mark(z) -{ 0 }-> 0 :|: z >= 0 mark(z) -{ 0 }-> 0 :|: z = 3 mark(z) -{ 0 }-> 1 + mark(z - 1) :|: z - 1 >= 0 mark(z) -{ 0 }-> 1 + mark(z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> 1 + 0 + 3 :|: z = 3 ---------------------------------------- (47) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: a__incr(v0) -> null_a__incr [0] a__adx(v0) -> null_a__adx [0] a__nats -> null_a__nats [0] a__zeros -> null_a__zeros [0] a__head(v0) -> null_a__head [0] a__tail(v0) -> null_a__tail [0] mark(v0) -> null_mark [0] A__INCR(v0) -> null_A__INCR [0] A__ADX(v0) -> null_A__ADX [0] A__HEAD(v0) -> null_A__HEAD [0] A__TAIL(v0) -> null_A__TAIL [0] MARK(v0) -> null_MARK [0] And the following fresh constants: null_a__incr, null_a__adx, null_a__nats, null_a__zeros, null_a__head, null_a__tail, null_mark, null_A__INCR, null_A__ADX, null_A__HEAD, null_A__TAIL, null_MARK, const ---------------------------------------- (48) 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__INCR(cons(z0, z1)) -> c1(MARK(z0)) [1] A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) [1] A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) [1] A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) [1] MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) [1] MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) [1] MARK(nats) -> c16(A__NATS) [1] MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) [1] MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) [1] MARK(cons(z0, z1)) -> c21(MARK(z0)) [1] MARK(s(z0)) -> c22(MARK(z0)) [1] A__NATS -> c6(A__ADX(a__zeros)) [1] a__incr(nil) -> nil [0] a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) [0] a__incr(z0) -> incr(z0) [0] a__adx(nil) -> nil [0] a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) [0] a__adx(z0) -> adx(z0) [0] a__nats -> a__adx(a__zeros) [0] a__nats -> nats [0] a__zeros -> cons(0, zeros) [0] a__zeros -> zeros [0] a__head(cons(z0, z1)) -> mark(z0) [0] a__head(z0) -> head(z0) [0] a__tail(cons(z0, z1)) -> mark(z1) [0] a__tail(z0) -> tail(z0) [0] mark(incr(z0)) -> a__incr(mark(z0)) [0] mark(adx(z0)) -> a__adx(mark(z0)) [0] mark(nats) -> a__nats [0] mark(zeros) -> a__zeros [0] mark(head(z0)) -> a__head(mark(z0)) [0] mark(tail(z0)) -> a__tail(mark(z0)) [0] mark(nil) -> nil [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(s(z0)) -> s(mark(z0)) [0] mark(0) -> 0 [0] a__incr(v0) -> null_a__incr [0] a__adx(v0) -> null_a__adx [0] a__nats -> null_a__nats [0] a__zeros -> null_a__zeros [0] a__head(v0) -> null_a__head [0] a__tail(v0) -> null_a__tail [0] mark(v0) -> null_mark [0] A__INCR(v0) -> null_A__INCR [0] A__ADX(v0) -> null_A__ADX [0] A__HEAD(v0) -> null_A__HEAD [0] A__TAIL(v0) -> null_A__TAIL [0] MARK(v0) -> null_MARK [0] The TRS has the following type information: A__INCR :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> c1:null_A__INCR cons :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c1 :: c14:c15:c16:c18:c19:c21:c22:null_MARK -> c1:null_A__INCR MARK :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> c14:c15:c16:c18:c19:c21:c22:null_MARK A__ADX :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> c4:null_A__ADX c4 :: c1:null_A__INCR -> c14:c15:c16:c18:c19:c21:c22:null_MARK -> c4:null_A__ADX mark :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark adx :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark A__HEAD :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> c10:null_A__HEAD c10 :: c14:c15:c16:c18:c19:c21:c22:null_MARK -> c10:null_A__HEAD A__TAIL :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> c12:null_A__TAIL c12 :: c14:c15:c16:c18:c19:c21:c22:null_MARK -> c12:null_A__TAIL incr :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c14 :: c1:null_A__INCR -> c14:c15:c16:c18:c19:c21:c22:null_MARK -> c14:c15:c16:c18:c19:c21:c22:null_MARK c15 :: c4:null_A__ADX -> c14:c15:c16:c18:c19:c21:c22:null_MARK -> c14:c15:c16:c18:c19:c21:c22:null_MARK nats :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c16 :: c6 -> c14:c15:c16:c18:c19:c21:c22:null_MARK A__NATS :: c6 head :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c18 :: c10:null_A__HEAD -> c14:c15:c16:c18:c19:c21:c22:null_MARK -> c14:c15:c16:c18:c19:c21:c22:null_MARK tail :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c19 :: c12:null_A__TAIL -> c14:c15:c16:c18:c19:c21:c22:null_MARK -> c14:c15:c16:c18:c19:c21:c22:null_MARK c21 :: c14:c15:c16:c18:c19:c21:c22:null_MARK -> c14:c15:c16:c18:c19:c21:c22:null_MARK s :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark c22 :: c14:c15:c16:c18:c19:c21:c22:null_MARK -> c14:c15:c16:c18:c19:c21:c22:null_MARK c6 :: c4:null_A__ADX -> c6 a__zeros :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark a__incr :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark nil :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark a__adx :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark a__nats :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark 0 :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark zeros :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark a__head :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark a__tail :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark -> cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__incr :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__adx :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__nats :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__zeros :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__head :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_a__tail :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_mark :: cons:adx:incr:nats:head:tail:s:nil:0:zeros:null_a__incr:null_a__adx:null_a__nats:null_a__zeros:null_a__head:null_a__tail:null_mark null_A__INCR :: c1:null_A__INCR null_A__ADX :: c4:null_A__ADX null_A__HEAD :: c10:null_A__HEAD null_A__TAIL :: c12:null_A__TAIL null_MARK :: c14:c15:c16:c18:c19:c21:c22:null_MARK const :: c6 Rewrite Strategy: INNERMOST ---------------------------------------- (49) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nats => 1 nil => 2 0 => 0 zeros => 3 null_a__incr => 0 null_a__adx => 0 null_a__nats => 0 null_a__zeros => 0 null_a__head => 0 null_a__tail => 0 null_mark => 0 null_A__INCR => 0 null_A__ADX => 0 null_A__HEAD => 0 null_A__TAIL => 0 null_MARK => 0 const => 0 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: A__ADX(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 A__ADX(z) -{ 1 }-> 1 + A__INCR(1 + mark(z0) + (1 + z1)) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__HEAD(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 A__HEAD(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__INCR(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 A__INCR(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__NATS -{ 1 }-> 1 + A__ADX(a__zeros) :|: A__TAIL(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 A__TAIL(z) -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__NATS :|: z = 1 MARK(z) -{ 1 }-> 1 + A__TAIL(mark(z0)) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__INCR(mark(z0)) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(mark(z0)) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__ADX(mark(z0)) + MARK(z0) :|: z = 1 + z0, z0 >= 0 a__adx(z) -{ 0 }-> a__incr(1 + mark(z0) + (1 + z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__adx(z) -{ 0 }-> 2 :|: z = 2 a__adx(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__adx(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__head(z) -{ 0 }-> mark(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__head(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__incr(z) -{ 0 }-> 2 :|: z = 2 a__incr(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__incr(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__incr(z) -{ 0 }-> 1 + (1 + mark(z0)) + (1 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__nats -{ 0 }-> a__adx(a__zeros) :|: a__nats -{ 0 }-> 1 :|: a__nats -{ 0 }-> 0 :|: a__tail(z) -{ 0 }-> mark(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__tail(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__zeros -{ 0 }-> 3 :|: a__zeros -{ 0 }-> 0 :|: a__zeros -{ 0 }-> 1 + 0 + 3 :|: mark(z) -{ 0 }-> a__zeros :|: z = 3 mark(z) -{ 0 }-> a__tail(mark(z0)) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__nats :|: z = 1 mark(z) -{ 0 }-> a__incr(mark(z0)) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__head(mark(z0)) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__adx(mark(z0)) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> 2 :|: z = 2 mark(z) -{ 0 }-> 0 :|: z = 0 mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 mark(z) -{ 0 }-> 1 + mark(z0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> 1 + mark(z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(z0)) -> c14(A__INCR(mark(z0)), MARK(z0)) by MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(zeros)) -> c14(A__INCR(a__zeros), MARK(zeros)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(nil)) -> c14(A__INCR(nil), MARK(nil)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(A__INCR(s(mark(z0))), MARK(s(z0))) MARK(incr(0)) -> c14(A__INCR(0), MARK(0)) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(zeros)) -> c14(A__INCR(a__zeros), MARK(zeros)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(nil)) -> c14(A__INCR(nil), MARK(nil)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(A__INCR(s(mark(z0))), MARK(s(z0))) MARK(incr(0)) -> c14(A__INCR(0), MARK(0)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(zeros)) -> c14(A__INCR(a__zeros), MARK(zeros)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(nil)) -> c14(A__INCR(nil), MARK(nil)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(A__INCR(s(mark(z0))), MARK(s(z0))) MARK(incr(0)) -> c14(A__INCR(0), MARK(0)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c15_2, c16_1, c18_2, c19_2, c21_1, c22_1, c6_1, c14_2 ---------------------------------------- (53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(incr(0)) -> c14(A__INCR(0), MARK(0)) MARK(incr(nil)) -> c14(A__INCR(nil), MARK(nil)) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(zeros)) -> c14(A__INCR(a__zeros), MARK(zeros)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(A__INCR(s(mark(z0))), MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(zeros)) -> c14(A__INCR(a__zeros), MARK(zeros)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(A__INCR(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c15_2, c16_1, c18_2, c19_2, c21_1, c22_1, c6_1, c14_2 ---------------------------------------- (55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c15_2, c16_1, c18_2, c19_2, c21_1, c22_1, c6_1, c14_2, c14_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(adx(z0)) -> c15(A__ADX(mark(z0)), MARK(z0)) by MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(zeros)) -> c15(A__ADX(a__zeros), MARK(zeros)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(nil)) -> c15(A__ADX(nil), MARK(nil)) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(A__ADX(s(mark(z0))), MARK(s(z0))) MARK(adx(0)) -> c15(A__ADX(0), MARK(0)) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(zeros)) -> c15(A__ADX(a__zeros), MARK(zeros)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(nil)) -> c15(A__ADX(nil), MARK(nil)) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(A__ADX(s(mark(z0))), MARK(s(z0))) MARK(adx(0)) -> c15(A__ADX(0), MARK(0)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(zeros)) -> c15(A__ADX(a__zeros), MARK(zeros)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(nil)) -> c15(A__ADX(nil), MARK(nil)) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(A__ADX(s(mark(z0))), MARK(s(z0))) MARK(adx(0)) -> c15(A__ADX(0), MARK(0)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c18_2, c19_2, c21_1, c22_1, c6_1, c14_2, c14_1, c15_2 ---------------------------------------- (59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(adx(nil)) -> c15(A__ADX(nil), MARK(nil)) MARK(adx(0)) -> c15(A__ADX(0), MARK(0)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(zeros)) -> c15(A__ADX(a__zeros), MARK(zeros)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(A__ADX(s(mark(z0))), MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(zeros)) -> c15(A__ADX(a__zeros), MARK(zeros)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(A__ADX(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c18_2, c19_2, c21_1, c22_1, c6_1, c14_2, c14_1, c15_2 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c18_2, c19_2, c21_1, c22_1, c6_1, c14_2, c14_1, c15_2, c15_1 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(z0)) -> c18(A__HEAD(mark(z0)), MARK(z0)) by MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(zeros)) -> c18(A__HEAD(a__zeros), MARK(zeros)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(nil)) -> c18(A__HEAD(nil), MARK(nil)) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(A__HEAD(s(mark(z0))), MARK(s(z0))) MARK(head(0)) -> c18(A__HEAD(0), MARK(0)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(zeros)) -> c18(A__HEAD(a__zeros), MARK(zeros)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(nil)) -> c18(A__HEAD(nil), MARK(nil)) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(A__HEAD(s(mark(z0))), MARK(s(z0))) MARK(head(0)) -> c18(A__HEAD(0), MARK(0)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(zeros)) -> c18(A__HEAD(a__zeros), MARK(zeros)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(nil)) -> c18(A__HEAD(nil), MARK(nil)) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(A__HEAD(s(mark(z0))), MARK(s(z0))) MARK(head(0)) -> c18(A__HEAD(0), MARK(0)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c19_2, c21_1, c22_1, c6_1, c14_2, c14_1, c15_2, c15_1, c18_2 ---------------------------------------- (65) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(head(nil)) -> c18(A__HEAD(nil), MARK(nil)) MARK(head(0)) -> c18(A__HEAD(0), MARK(0)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(zeros)) -> c18(A__HEAD(a__zeros), MARK(zeros)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(A__HEAD(s(mark(z0))), MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(zeros)) -> c18(A__HEAD(a__zeros), MARK(zeros)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(A__HEAD(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c19_2, c21_1, c22_1, c6_1, c14_2, c14_1, c15_2, c15_1, c18_2 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c19_2, c21_1, c22_1, c6_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(z0)) -> c19(A__TAIL(mark(z0)), MARK(z0)) by MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros), MARK(zeros)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(nil)) -> c19(A__TAIL(nil), MARK(nil)) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(A__TAIL(s(mark(z0))), MARK(s(z0))) MARK(tail(0)) -> c19(A__TAIL(0), MARK(0)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros), MARK(zeros)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(nil)) -> c19(A__TAIL(nil), MARK(nil)) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(A__TAIL(s(mark(z0))), MARK(s(z0))) MARK(tail(0)) -> c19(A__TAIL(0), MARK(0)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros), MARK(zeros)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(nil)) -> c19(A__TAIL(nil), MARK(nil)) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(A__TAIL(s(mark(z0))), MARK(s(z0))) MARK(tail(0)) -> c19(A__TAIL(0), MARK(0)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c6_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2 ---------------------------------------- (71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(tail(0)) -> c19(A__TAIL(0), MARK(0)) MARK(tail(nil)) -> c19(A__TAIL(nil), MARK(nil)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros), MARK(zeros)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(A__TAIL(s(mark(z0))), MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros), MARK(zeros)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(A__TAIL(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c6_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) A__NATS -> c6(A__ADX(a__zeros)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c6_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__NATS -> c6(A__ADX(a__zeros)) by A__NATS -> c6(A__ADX(cons(0, zeros))) A__NATS -> c6(A__ADX(zeros)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) A__NATS -> c6(A__ADX(zeros)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) A__NATS -> c6(A__ADX(zeros)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__NATS -> c6(A__ADX(zeros)) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(incr(z0))) -> c14(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) by MARK(incr(incr(x0))) -> c14(A__INCR(incr(mark(x0))), MARK(incr(x0))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil)), MARK(incr(nil))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0)), MARK(incr(0))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(x0))) -> c14(A__INCR(incr(mark(x0))), MARK(incr(x0))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil)), MARK(incr(nil))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0)), MARK(incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(x0))) -> c14(A__INCR(incr(mark(x0))), MARK(incr(x0))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil)), MARK(incr(nil))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0)), MARK(incr(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(adx(z0))) -> c14(A__INCR(a__adx(mark(z0))), MARK(adx(z0))) by MARK(incr(adx(x0))) -> c14(A__INCR(adx(mark(x0))), MARK(adx(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil)), MARK(adx(nil))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0)), MARK(adx(0))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(x0))) -> c14(A__INCR(adx(mark(x0))), MARK(adx(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil)), MARK(adx(nil))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0)), MARK(adx(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(x0))) -> c14(A__INCR(adx(mark(x0))), MARK(adx(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil)), MARK(adx(nil))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0)), MARK(adx(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (85) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(nats)) -> c14(A__INCR(a__nats), MARK(nats)) by MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(A__INCR(nats), MARK(nats)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(A__INCR(nats), MARK(nats)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(A__INCR(nats), MARK(nats)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (89) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(head(z0))) -> c14(A__INCR(a__head(mark(z0))), MARK(head(z0))) by MARK(incr(head(x0))) -> c14(A__INCR(head(mark(x0))), MARK(head(x0))) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil)), MARK(head(nil))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0)), MARK(head(0))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(x0))) -> c14(A__INCR(head(mark(x0))), MARK(head(x0))) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil)), MARK(head(nil))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0)), MARK(head(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(x0))) -> c14(A__INCR(head(mark(x0))), MARK(head(x0))) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil)), MARK(head(nil))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0)), MARK(head(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(tail(z0))) -> c14(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) by MARK(incr(tail(x0))) -> c14(A__INCR(tail(mark(x0))), MARK(tail(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil)), MARK(tail(nil))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0)), MARK(tail(0))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(x0))) -> c14(A__INCR(tail(mark(x0))), MARK(tail(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil)), MARK(tail(nil))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0)), MARK(tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(x0))) -> c14(A__INCR(tail(mark(x0))), MARK(tail(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil)), MARK(tail(nil))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0)), MARK(tail(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (97) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(zeros)) -> c14(A__INCR(a__zeros)) by MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(incr(zeros)) -> c14(A__INCR(zeros)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(incr(zeros)) -> c14(A__INCR(zeros)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(incr(zeros)) -> c14(A__INCR(zeros)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (101) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(zeros)) -> c14(A__INCR(zeros)) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(adx(incr(z0))) -> c15(A__ADX(a__incr(mark(z0))), MARK(incr(z0))) by MARK(adx(incr(x0))) -> c15(A__ADX(incr(mark(x0))), MARK(incr(x0))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil)), MARK(incr(nil))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0)), MARK(incr(0))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(x0))) -> c15(A__ADX(incr(mark(x0))), MARK(incr(x0))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil)), MARK(incr(nil))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0)), MARK(incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(x0))) -> c15(A__ADX(incr(mark(x0))), MARK(incr(x0))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil)), MARK(incr(nil))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0)), MARK(incr(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(adx(adx(z0))) -> c15(A__ADX(a__adx(mark(z0))), MARK(adx(z0))) by MARK(adx(adx(x0))) -> c15(A__ADX(adx(mark(x0))), MARK(adx(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil)), MARK(adx(nil))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0)), MARK(adx(0))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(x0))) -> c15(A__ADX(adx(mark(x0))), MARK(adx(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil)), MARK(adx(nil))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0)), MARK(adx(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(x0))) -> c15(A__ADX(adx(mark(x0))), MARK(adx(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil)), MARK(adx(nil))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0)), MARK(adx(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (109) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(adx(nats)) -> c15(A__ADX(a__nats), MARK(nats)) by MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(A__ADX(nats), MARK(nats)) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(A__ADX(nats), MARK(nats)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(A__ADX(nats), MARK(nats)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(adx(head(z0))) -> c15(A__ADX(a__head(mark(z0))), MARK(head(z0))) by MARK(adx(head(x0))) -> c15(A__ADX(head(mark(x0))), MARK(head(x0))) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil)), MARK(head(nil))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0)), MARK(head(0))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(x0))) -> c15(A__ADX(head(mark(x0))), MARK(head(x0))) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil)), MARK(head(nil))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0)), MARK(head(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(x0))) -> c15(A__ADX(head(mark(x0))), MARK(head(x0))) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil)), MARK(head(nil))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0)), MARK(head(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(adx(tail(z0))) -> c15(A__ADX(a__tail(mark(z0))), MARK(tail(z0))) by MARK(adx(tail(x0))) -> c15(A__ADX(tail(mark(x0))), MARK(tail(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil)), MARK(tail(nil))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0)), MARK(tail(0))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(x0))) -> c15(A__ADX(tail(mark(x0))), MARK(tail(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil)), MARK(tail(nil))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0)), MARK(tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(x0))) -> c15(A__ADX(tail(mark(x0))), MARK(tail(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil)), MARK(tail(nil))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0)), MARK(tail(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (121) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(adx(zeros)) -> c15(A__ADX(a__zeros)) by MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(adx(zeros)) -> c15(A__ADX(zeros)) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(adx(zeros)) -> c15(A__ADX(zeros)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(adx(zeros)) -> c15(A__ADX(zeros)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (125) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(adx(zeros)) -> c15(A__ADX(zeros)) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(incr(z0))) -> c18(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) by MARK(head(incr(x0))) -> c18(A__HEAD(incr(mark(x0))), MARK(incr(x0))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil)), MARK(incr(nil))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0)), MARK(incr(0))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(x0))) -> c18(A__HEAD(incr(mark(x0))), MARK(incr(x0))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil)), MARK(incr(nil))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0)), MARK(incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(x0))) -> c18(A__HEAD(incr(mark(x0))), MARK(incr(x0))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil)), MARK(incr(nil))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0)), MARK(incr(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (129) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(adx(z0))) -> c18(A__HEAD(a__adx(mark(z0))), MARK(adx(z0))) by MARK(head(adx(x0))) -> c18(A__HEAD(adx(mark(x0))), MARK(adx(x0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil)), MARK(adx(nil))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0)), MARK(adx(0))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(x0))) -> c18(A__HEAD(adx(mark(x0))), MARK(adx(x0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil)), MARK(adx(nil))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0)), MARK(adx(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(x0))) -> c18(A__HEAD(adx(mark(x0))), MARK(adx(x0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil)), MARK(adx(nil))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0)), MARK(adx(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (133) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(nats)) -> c18(A__HEAD(a__nats), MARK(nats)) by MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(A__HEAD(nats), MARK(nats)) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(A__HEAD(nats), MARK(nats)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(A__HEAD(nats), MARK(nats)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (137) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(head(z0))) -> c18(A__HEAD(a__head(mark(z0))), MARK(head(z0))) by MARK(head(head(x0))) -> c18(A__HEAD(head(mark(x0))), MARK(head(x0))) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil)), MARK(head(nil))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0)), MARK(head(0))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(x0))) -> c18(A__HEAD(head(mark(x0))), MARK(head(x0))) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil)), MARK(head(nil))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0)), MARK(head(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(x0))) -> c18(A__HEAD(head(mark(x0))), MARK(head(x0))) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil)), MARK(head(nil))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0)), MARK(head(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (141) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(tail(z0))) -> c18(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) by MARK(head(tail(x0))) -> c18(A__HEAD(tail(mark(x0))), MARK(tail(x0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil)), MARK(tail(nil))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0)), MARK(tail(0))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(x0))) -> c18(A__HEAD(tail(mark(x0))), MARK(tail(x0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil)), MARK(tail(nil))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0)), MARK(tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(x0))) -> c18(A__HEAD(tail(mark(x0))), MARK(tail(x0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil)), MARK(tail(nil))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0)), MARK(tail(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (145) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (147) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(zeros)) -> c18(A__HEAD(a__zeros)) by MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(head(zeros)) -> c18(A__HEAD(zeros)) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(head(zeros)) -> c18(A__HEAD(zeros)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(head(zeros)) -> c18(A__HEAD(zeros)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (149) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(head(zeros)) -> c18(A__HEAD(zeros)) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (151) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(incr(z0))) -> c19(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) by MARK(tail(incr(x0))) -> c19(A__TAIL(incr(mark(x0))), MARK(incr(x0))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil)), MARK(incr(nil))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0)), MARK(incr(0))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(x0))) -> c19(A__TAIL(incr(mark(x0))), MARK(incr(x0))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil)), MARK(incr(nil))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0)), MARK(incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(x0))) -> c19(A__TAIL(incr(mark(x0))), MARK(incr(x0))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil)), MARK(incr(nil))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0)), MARK(incr(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (153) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (155) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(adx(z0))) -> c19(A__TAIL(a__adx(mark(z0))), MARK(adx(z0))) by MARK(tail(adx(x0))) -> c19(A__TAIL(adx(mark(x0))), MARK(adx(x0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil)), MARK(adx(nil))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0)), MARK(adx(0))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(x0))) -> c19(A__TAIL(adx(mark(x0))), MARK(adx(x0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil)), MARK(adx(nil))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0)), MARK(adx(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(x0))) -> c19(A__TAIL(adx(mark(x0))), MARK(adx(x0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil)), MARK(adx(nil))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0)), MARK(adx(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (157) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (159) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(nats)) -> c19(A__TAIL(a__nats), MARK(nats)) by MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(A__TAIL(nats), MARK(nats)) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(A__TAIL(nats), MARK(nats)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(A__TAIL(nats), MARK(nats)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (161) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (163) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(head(z0))) -> c19(A__TAIL(a__head(mark(z0))), MARK(head(z0))) by MARK(tail(head(x0))) -> c19(A__TAIL(head(mark(x0))), MARK(head(x0))) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil)), MARK(head(nil))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0)), MARK(head(0))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(x0))) -> c19(A__TAIL(head(mark(x0))), MARK(head(x0))) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil)), MARK(head(nil))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0)), MARK(head(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(x0))) -> c19(A__TAIL(head(mark(x0))), MARK(head(x0))) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil)), MARK(head(nil))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0)), MARK(head(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (165) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (167) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(tail(z0))) -> c19(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) by MARK(tail(tail(x0))) -> c19(A__TAIL(tail(mark(x0))), MARK(tail(x0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil)), MARK(tail(nil))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0)), MARK(tail(0))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(x0))) -> c19(A__TAIL(tail(mark(x0))), MARK(tail(x0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil)), MARK(tail(nil))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0)), MARK(tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(x0))) -> c19(A__TAIL(tail(mark(x0))), MARK(tail(x0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil)), MARK(tail(nil))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0)), MARK(tail(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (169) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (171) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(zeros)) -> c19(A__TAIL(a__zeros)) by MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(tail(zeros)) -> c19(A__TAIL(zeros)) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(tail(zeros)) -> c19(A__TAIL(zeros)) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(tail(zeros)) -> c19(A__TAIL(zeros)) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (173) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(tail(zeros)) -> c19(A__TAIL(zeros)) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (175) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(incr(nil))) -> c14(A__INCR(a__incr(nil))) by MARK(incr(incr(nil))) -> c14(A__INCR(nil)) MARK(incr(incr(nil))) -> c14(A__INCR(incr(nil))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(incr(nil))) -> c14(A__INCR(nil)) MARK(incr(incr(nil))) -> c14(A__INCR(incr(nil))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(incr(nil))) -> c14(A__INCR(nil)) MARK(incr(incr(nil))) -> c14(A__INCR(incr(nil))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (177) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(incr(incr(nil))) -> c14(A__INCR(nil)) MARK(incr(incr(nil))) -> c14(A__INCR(incr(nil))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (179) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(incr(0))) -> c14(A__INCR(a__incr(0))) by MARK(incr(incr(0))) -> c14(A__INCR(incr(0))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(incr(0))) -> c14(A__INCR(incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(incr(0))) -> c14(A__INCR(incr(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (181) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(incr(0))) -> c14(A__INCR(incr(0))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (183) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(adx(nil))) -> c14(A__INCR(a__adx(nil))) by MARK(incr(adx(nil))) -> c14(A__INCR(nil)) MARK(incr(adx(nil))) -> c14(A__INCR(adx(nil))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(adx(nil))) -> c14(A__INCR(nil)) MARK(incr(adx(nil))) -> c14(A__INCR(adx(nil))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(adx(nil))) -> c14(A__INCR(nil)) MARK(incr(adx(nil))) -> c14(A__INCR(adx(nil))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (185) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(incr(adx(nil))) -> c14(A__INCR(adx(nil))) MARK(incr(adx(nil))) -> c14(A__INCR(nil)) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (187) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(adx(0))) -> c14(A__INCR(a__adx(0))) by MARK(incr(adx(0))) -> c14(A__INCR(adx(0))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(adx(0))) -> c14(A__INCR(adx(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(adx(0))) -> c14(A__INCR(adx(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (189) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(adx(0))) -> c14(A__INCR(adx(0))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (191) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(head(nil))) -> c14(A__INCR(a__head(nil))) by MARK(incr(head(nil))) -> c14(A__INCR(head(nil))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(head(nil))) -> c14(A__INCR(head(nil))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(head(nil))) -> c14(A__INCR(head(nil))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (193) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(head(nil))) -> c14(A__INCR(head(nil))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (195) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(head(0))) -> c14(A__INCR(a__head(0))) by MARK(incr(head(0))) -> c14(A__INCR(head(0))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(head(0))) -> c14(A__INCR(head(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(head(0))) -> c14(A__INCR(head(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (197) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(head(0))) -> c14(A__INCR(head(0))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (199) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(tail(nil))) -> c14(A__INCR(a__tail(nil))) by MARK(incr(tail(nil))) -> c14(A__INCR(tail(nil))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(tail(nil))) -> c14(A__INCR(tail(nil))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(tail(nil))) -> c14(A__INCR(tail(nil))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (201) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(tail(nil))) -> c14(A__INCR(tail(nil))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (203) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(tail(0))) -> c14(A__INCR(a__tail(0))) by MARK(incr(tail(0))) -> c14(A__INCR(tail(0))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(tail(0))) -> c14(A__INCR(tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(incr(tail(0))) -> c14(A__INCR(tail(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (205) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(tail(0))) -> c14(A__INCR(tail(0))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (207) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(adx(incr(nil))) -> c15(A__ADX(a__incr(nil))) by MARK(adx(incr(nil))) -> c15(A__ADX(nil)) MARK(adx(incr(nil))) -> c15(A__ADX(incr(nil))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(incr(nil))) -> c15(A__ADX(nil)) MARK(adx(incr(nil))) -> c15(A__ADX(incr(nil))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(incr(nil))) -> c15(A__ADX(nil)) MARK(adx(incr(nil))) -> c15(A__ADX(incr(nil))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (209) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(adx(incr(nil))) -> c15(A__ADX(nil)) MARK(adx(incr(nil))) -> c15(A__ADX(incr(nil))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (211) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(adx(incr(0))) -> c15(A__ADX(a__incr(0))) by MARK(adx(incr(0))) -> c15(A__ADX(incr(0))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(incr(0))) -> c15(A__ADX(incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(incr(0))) -> c15(A__ADX(incr(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (213) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(adx(incr(0))) -> c15(A__ADX(incr(0))) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (215) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(adx(adx(nil))) -> c15(A__ADX(a__adx(nil))) by MARK(adx(adx(nil))) -> c15(A__ADX(nil)) MARK(adx(adx(nil))) -> c15(A__ADX(adx(nil))) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(adx(nil))) -> c15(A__ADX(nil)) MARK(adx(adx(nil))) -> c15(A__ADX(adx(nil))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(adx(nil))) -> c15(A__ADX(nil)) MARK(adx(adx(nil))) -> c15(A__ADX(adx(nil))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (217) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(adx(adx(nil))) -> c15(A__ADX(adx(nil))) MARK(adx(adx(nil))) -> c15(A__ADX(nil)) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (219) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(adx(adx(0))) -> c15(A__ADX(a__adx(0))) by MARK(adx(adx(0))) -> c15(A__ADX(adx(0))) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(adx(0))) -> c15(A__ADX(adx(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(adx(0))) -> c15(A__ADX(adx(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (221) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(adx(adx(0))) -> c15(A__ADX(adx(0))) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (223) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(adx(head(nil))) -> c15(A__ADX(a__head(nil))) by MARK(adx(head(nil))) -> c15(A__ADX(head(nil))) ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(head(nil))) -> c15(A__ADX(head(nil))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(head(nil))) -> c15(A__ADX(head(nil))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (225) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(adx(head(nil))) -> c15(A__ADX(head(nil))) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (227) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(adx(head(0))) -> c15(A__ADX(a__head(0))) by MARK(adx(head(0))) -> c15(A__ADX(head(0))) ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(head(0))) -> c15(A__ADX(head(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(head(0))) -> c15(A__ADX(head(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (229) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(adx(head(0))) -> c15(A__ADX(head(0))) ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (231) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(adx(tail(nil))) -> c15(A__ADX(a__tail(nil))) by MARK(adx(tail(nil))) -> c15(A__ADX(tail(nil))) ---------------------------------------- (232) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(tail(nil))) -> c15(A__ADX(tail(nil))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(tail(nil))) -> c15(A__ADX(tail(nil))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (233) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(adx(tail(nil))) -> c15(A__ADX(tail(nil))) ---------------------------------------- (234) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (235) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(adx(tail(0))) -> c15(A__ADX(a__tail(0))) by MARK(adx(tail(0))) -> c15(A__ADX(tail(0))) ---------------------------------------- (236) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(tail(0))) -> c15(A__ADX(tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(adx(tail(0))) -> c15(A__ADX(tail(0))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (237) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(adx(tail(0))) -> c15(A__ADX(tail(0))) ---------------------------------------- (238) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (239) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(incr(nil))) -> c18(A__HEAD(a__incr(nil))) by MARK(head(incr(nil))) -> c18(A__HEAD(nil)) MARK(head(incr(nil))) -> c18(A__HEAD(incr(nil))) ---------------------------------------- (240) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(head(incr(nil))) -> c18(A__HEAD(nil)) MARK(head(incr(nil))) -> c18(A__HEAD(incr(nil))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) MARK(head(incr(nil))) -> c18(A__HEAD(nil)) MARK(head(incr(nil))) -> c18(A__HEAD(incr(nil))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (241) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(head(incr(nil))) -> c18(A__HEAD(incr(nil))) MARK(head(incr(nil))) -> c18(A__HEAD(nil)) ---------------------------------------- (242) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) a__adx(nil) -> nil a__adx(cons(z0, z1)) -> a__incr(cons(mark(z0), adx(z1))) a__adx(z0) -> adx(z0) a__nats -> a__adx(a__zeros) a__nats -> nats a__zeros -> cons(0, zeros) a__zeros -> zeros a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) mark(incr(z0)) -> a__incr(mark(z0)) mark(adx(z0)) -> a__adx(mark(z0)) mark(nats) -> a__nats mark(zeros) -> a__zeros mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(nil) -> nil mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 Tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) S tuples: A__INCR(cons(z0, z1)) -> c1(MARK(z0)) A__ADX(cons(z0, z1)) -> c4(A__INCR(cons(mark(z0), adx(z1))), MARK(z0)) A__HEAD(cons(z0, z1)) -> c10(MARK(z0)) A__TAIL(cons(z0, z1)) -> c12(MARK(z1)) MARK(nats) -> c16(A__NATS) MARK(cons(z0, z1)) -> c21(MARK(z0)) MARK(s(z0)) -> c22(MARK(z0)) MARK(incr(cons(z0, z1))) -> c14(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c14(MARK(s(z0))) MARK(adx(cons(z0, z1))) -> c15(A__ADX(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(adx(s(z0))) -> c15(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c18(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c18(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c19(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c19(MARK(s(z0))) A__NATS -> c6(A__ADX(cons(0, zeros))) MARK(incr(incr(incr(z0)))) -> c14(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(adx(z0)))) -> c14(A__INCR(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(incr(incr(nats))) -> c14(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(zeros))) -> c14(A__INCR(a__incr(a__zeros)), MARK(incr(zeros))) MARK(incr(incr(head(z0)))) -> c14(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c14(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c14(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c14(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c14(MARK(incr(x0))) MARK(incr(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> c14(A__INCR(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(incr(adx(nats))) -> c14(A__INCR(a__adx(a__nats)), MARK(adx(nats))) MARK(incr(adx(zeros))) -> c14(A__INCR(a__adx(a__zeros)), MARK(adx(zeros))) MARK(incr(adx(head(z0)))) -> c14(A__INCR(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(incr(adx(tail(z0)))) -> c14(A__INCR(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(incr(adx(cons(z0, z1)))) -> c14(A__INCR(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(incr(adx(s(z0)))) -> c14(A__INCR(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(incr(adx(x0))) -> c14(MARK(adx(x0))) MARK(incr(nats)) -> c14(A__INCR(a__adx(a__zeros)), MARK(nats)) MARK(incr(nats)) -> c14(MARK(nats)) MARK(incr(head(incr(z0)))) -> c14(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(adx(z0)))) -> c14(A__INCR(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(incr(head(nats))) -> c14(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(zeros))) -> c14(A__INCR(a__head(a__zeros)), MARK(head(zeros))) MARK(incr(head(head(z0)))) -> c14(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c14(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c14(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c14(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c14(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c14(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(adx(z0)))) -> c14(A__INCR(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(incr(tail(nats))) -> c14(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(zeros))) -> c14(A__INCR(a__tail(a__zeros)), MARK(tail(zeros))) MARK(incr(tail(head(z0)))) -> c14(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c14(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c14(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c14(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c14(MARK(tail(x0))) MARK(incr(zeros)) -> c14(A__INCR(cons(0, zeros))) MARK(adx(incr(incr(z0)))) -> c15(A__ADX(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(adx(incr(adx(z0)))) -> c15(A__ADX(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(adx(incr(nats))) -> c15(A__ADX(a__incr(a__nats)), MARK(incr(nats))) MARK(adx(incr(zeros))) -> c15(A__ADX(a__incr(a__zeros)), MARK(incr(zeros))) MARK(adx(incr(head(z0)))) -> c15(A__ADX(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(adx(incr(tail(z0)))) -> c15(A__ADX(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(adx(incr(cons(z0, z1)))) -> c15(A__ADX(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(adx(incr(s(z0)))) -> c15(A__ADX(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(adx(incr(x0))) -> c15(MARK(incr(x0))) MARK(adx(adx(incr(z0)))) -> c15(A__ADX(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(adx(adx(adx(z0)))) -> c15(A__ADX(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(adx(adx(nats))) -> c15(A__ADX(a__adx(a__nats)), MARK(adx(nats))) MARK(adx(adx(zeros))) -> c15(A__ADX(a__adx(a__zeros)), MARK(adx(zeros))) MARK(adx(adx(head(z0)))) -> c15(A__ADX(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(adx(adx(tail(z0)))) -> c15(A__ADX(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(adx(adx(cons(z0, z1)))) -> c15(A__ADX(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(adx(adx(s(z0)))) -> c15(A__ADX(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(adx(adx(x0))) -> c15(MARK(adx(x0))) MARK(adx(nats)) -> c15(A__ADX(a__adx(a__zeros)), MARK(nats)) MARK(adx(nats)) -> c15(MARK(nats)) MARK(adx(head(incr(z0)))) -> c15(A__ADX(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(adx(head(adx(z0)))) -> c15(A__ADX(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(adx(head(nats))) -> c15(A__ADX(a__head(a__nats)), MARK(head(nats))) MARK(adx(head(zeros))) -> c15(A__ADX(a__head(a__zeros)), MARK(head(zeros))) MARK(adx(head(head(z0)))) -> c15(A__ADX(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(adx(head(tail(z0)))) -> c15(A__ADX(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(adx(head(cons(z0, z1)))) -> c15(A__ADX(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(adx(head(s(z0)))) -> c15(A__ADX(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(adx(head(x0))) -> c15(MARK(head(x0))) MARK(adx(tail(incr(z0)))) -> c15(A__ADX(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(adx(tail(adx(z0)))) -> c15(A__ADX(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(adx(tail(nats))) -> c15(A__ADX(a__tail(a__nats)), MARK(tail(nats))) MARK(adx(tail(zeros))) -> c15(A__ADX(a__tail(a__zeros)), MARK(tail(zeros))) MARK(adx(tail(head(z0)))) -> c15(A__ADX(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(adx(tail(tail(z0)))) -> c15(A__ADX(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(adx(tail(cons(z0, z1)))) -> c15(A__ADX(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(adx(tail(s(z0)))) -> c15(A__ADX(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(adx(tail(x0))) -> c15(MARK(tail(x0))) MARK(adx(zeros)) -> c15(A__ADX(cons(0, zeros))) MARK(head(incr(incr(z0)))) -> c18(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(adx(z0)))) -> c18(A__HEAD(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(head(incr(nats))) -> c18(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(zeros))) -> c18(A__HEAD(a__incr(a__zeros)), MARK(incr(zeros))) MARK(head(incr(head(z0)))) -> c18(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c18(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c18(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c18(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c18(MARK(incr(x0))) MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) MARK(head(adx(incr(z0)))) -> c18(A__HEAD(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(head(adx(adx(z0)))) -> c18(A__HEAD(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(head(adx(nats))) -> c18(A__HEAD(a__adx(a__nats)), MARK(adx(nats))) MARK(head(adx(zeros))) -> c18(A__HEAD(a__adx(a__zeros)), MARK(adx(zeros))) MARK(head(adx(head(z0)))) -> c18(A__HEAD(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(head(adx(tail(z0)))) -> c18(A__HEAD(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(head(adx(cons(z0, z1)))) -> c18(A__HEAD(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(head(adx(s(z0)))) -> c18(A__HEAD(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(head(adx(x0))) -> c18(MARK(adx(x0))) MARK(head(adx(nil))) -> c18(A__HEAD(a__adx(nil))) MARK(head(adx(0))) -> c18(A__HEAD(a__adx(0))) MARK(head(nats)) -> c18(A__HEAD(a__adx(a__zeros)), MARK(nats)) MARK(head(nats)) -> c18(MARK(nats)) MARK(head(head(incr(z0)))) -> c18(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(adx(z0)))) -> c18(A__HEAD(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(head(head(nats))) -> c18(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(zeros))) -> c18(A__HEAD(a__head(a__zeros)), MARK(head(zeros))) MARK(head(head(head(z0)))) -> c18(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c18(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c18(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c18(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c18(MARK(head(x0))) MARK(head(head(nil))) -> c18(A__HEAD(a__head(nil))) MARK(head(head(0))) -> c18(A__HEAD(a__head(0))) MARK(head(tail(incr(z0)))) -> c18(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(adx(z0)))) -> c18(A__HEAD(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(head(tail(nats))) -> c18(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(zeros))) -> c18(A__HEAD(a__tail(a__zeros)), MARK(tail(zeros))) MARK(head(tail(head(z0)))) -> c18(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c18(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c18(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c18(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c18(MARK(tail(x0))) MARK(head(tail(nil))) -> c18(A__HEAD(a__tail(nil))) MARK(head(tail(0))) -> c18(A__HEAD(a__tail(0))) MARK(head(zeros)) -> c18(A__HEAD(cons(0, zeros))) MARK(tail(incr(incr(z0)))) -> c19(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(adx(z0)))) -> c19(A__TAIL(a__incr(a__adx(mark(z0)))), MARK(incr(adx(z0)))) MARK(tail(incr(nats))) -> c19(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(zeros))) -> c19(A__TAIL(a__incr(a__zeros)), MARK(incr(zeros))) MARK(tail(incr(head(z0)))) -> c19(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c19(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c19(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c19(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c19(MARK(incr(x0))) MARK(tail(incr(nil))) -> c19(A__TAIL(a__incr(nil))) MARK(tail(incr(0))) -> c19(A__TAIL(a__incr(0))) MARK(tail(adx(incr(z0)))) -> c19(A__TAIL(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(tail(adx(adx(z0)))) -> c19(A__TAIL(a__adx(a__adx(mark(z0)))), MARK(adx(adx(z0)))) MARK(tail(adx(nats))) -> c19(A__TAIL(a__adx(a__nats)), MARK(adx(nats))) MARK(tail(adx(zeros))) -> c19(A__TAIL(a__adx(a__zeros)), MARK(adx(zeros))) MARK(tail(adx(head(z0)))) -> c19(A__TAIL(a__adx(a__head(mark(z0)))), MARK(adx(head(z0)))) MARK(tail(adx(tail(z0)))) -> c19(A__TAIL(a__adx(a__tail(mark(z0)))), MARK(adx(tail(z0)))) MARK(tail(adx(cons(z0, z1)))) -> c19(A__TAIL(a__adx(cons(mark(z0), z1))), MARK(adx(cons(z0, z1)))) MARK(tail(adx(s(z0)))) -> c19(A__TAIL(a__adx(s(mark(z0)))), MARK(adx(s(z0)))) MARK(tail(adx(x0))) -> c19(MARK(adx(x0))) MARK(tail(adx(nil))) -> c19(A__TAIL(a__adx(nil))) MARK(tail(adx(0))) -> c19(A__TAIL(a__adx(0))) MARK(tail(nats)) -> c19(A__TAIL(a__adx(a__zeros)), MARK(nats)) MARK(tail(nats)) -> c19(MARK(nats)) MARK(tail(head(incr(z0)))) -> c19(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(adx(z0)))) -> c19(A__TAIL(a__head(a__adx(mark(z0)))), MARK(head(adx(z0)))) MARK(tail(head(nats))) -> c19(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(zeros))) -> c19(A__TAIL(a__head(a__zeros)), MARK(head(zeros))) MARK(tail(head(head(z0)))) -> c19(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c19(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c19(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c19(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c19(MARK(head(x0))) MARK(tail(head(nil))) -> c19(A__TAIL(a__head(nil))) MARK(tail(head(0))) -> c19(A__TAIL(a__head(0))) MARK(tail(tail(incr(z0)))) -> c19(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(adx(z0)))) -> c19(A__TAIL(a__tail(a__adx(mark(z0)))), MARK(tail(adx(z0)))) MARK(tail(tail(nats))) -> c19(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(zeros))) -> c19(A__TAIL(a__tail(a__zeros)), MARK(tail(zeros))) MARK(tail(tail(head(z0)))) -> c19(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c19(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c19(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c19(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c19(MARK(tail(x0))) MARK(tail(tail(nil))) -> c19(A__TAIL(a__tail(nil))) MARK(tail(tail(0))) -> c19(A__TAIL(a__tail(0))) MARK(tail(zeros)) -> c19(A__TAIL(cons(0, zeros))) K tuples:none Defined Rule Symbols: a__incr_1, a__adx_1, a__nats, a__zeros, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__ADX_1, A__HEAD_1, A__TAIL_1, MARK_1, A__NATS Compound Symbols: c1_1, c4_2, c10_1, c12_1, c16_1, c21_1, c22_1, c14_2, c14_1, c15_2, c15_1, c18_2, c18_1, c19_2, c19_1, c6_1 ---------------------------------------- (243) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (244) 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__incr(nil) -> nil [1] a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L)) [1] a__adx(nil) -> nil [1] a__adx(cons(X, L)) -> a__incr(cons(mark(X), adx(L))) [1] a__nats -> a__adx(a__zeros) [1] a__zeros -> cons(0, zeros) [1] a__head(cons(X, L)) -> mark(X) [1] a__tail(cons(X, L)) -> mark(L) [1] mark(incr(X)) -> a__incr(mark(X)) [1] mark(adx(X)) -> a__adx(mark(X)) [1] mark(nats) -> a__nats [1] mark(zeros) -> a__zeros [1] mark(head(X)) -> a__head(mark(X)) [1] mark(tail(X)) -> a__tail(mark(X)) [1] mark(nil) -> nil [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(s(X)) -> s(mark(X)) [1] mark(0) -> 0 [1] a__incr(X) -> incr(X) [1] a__adx(X) -> adx(X) [1] a__nats -> nats [1] a__zeros -> zeros [1] a__head(X) -> head(X) [1] a__tail(X) -> tail(X) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (245) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (246) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a__incr(nil) -> nil [1] a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L)) [1] a__adx(nil) -> nil [1] a__adx(cons(X, L)) -> a__incr(cons(mark(X), adx(L))) [1] a__nats -> a__adx(a__zeros) [1] a__zeros -> cons(0, zeros) [1] a__head(cons(X, L)) -> mark(X) [1] a__tail(cons(X, L)) -> mark(L) [1] mark(incr(X)) -> a__incr(mark(X)) [1] mark(adx(X)) -> a__adx(mark(X)) [1] mark(nats) -> a__nats [1] mark(zeros) -> a__zeros [1] mark(head(X)) -> a__head(mark(X)) [1] mark(tail(X)) -> a__tail(mark(X)) [1] mark(nil) -> nil [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(s(X)) -> s(mark(X)) [1] mark(0) -> 0 [1] a__incr(X) -> incr(X) [1] a__adx(X) -> adx(X) [1] a__nats -> nats [1] a__zeros -> zeros [1] a__head(X) -> head(X) [1] a__tail(X) -> tail(X) [1] The TRS has the following type information: a__incr :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail nil :: nil:cons:s:incr:adx:0:zeros:nats:head:tail cons :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail s :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail mark :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail incr :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail a__adx :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail adx :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail a__nats :: nil:cons:s:incr:adx:0:zeros:nats:head:tail a__zeros :: nil:cons:s:incr:adx:0:zeros:nats:head:tail 0 :: nil:cons:s:incr:adx:0:zeros:nats:head:tail zeros :: nil:cons:s:incr:adx:0:zeros:nats:head:tail a__head :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail a__tail :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail nats :: nil:cons:s:incr:adx:0:zeros:nats:head:tail head :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail tail :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail Rewrite Strategy: INNERMOST ---------------------------------------- (247) 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 ---------------------------------------- (248) 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__incr(nil) -> nil [1] a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L)) [1] a__adx(nil) -> nil [1] a__adx(cons(X, L)) -> a__incr(cons(mark(X), adx(L))) [1] a__nats -> a__adx(a__zeros) [1] a__zeros -> cons(0, zeros) [1] a__head(cons(X, L)) -> mark(X) [1] a__tail(cons(X, L)) -> mark(L) [1] mark(incr(X)) -> a__incr(mark(X)) [1] mark(adx(X)) -> a__adx(mark(X)) [1] mark(nats) -> a__nats [1] mark(zeros) -> a__zeros [1] mark(head(X)) -> a__head(mark(X)) [1] mark(tail(X)) -> a__tail(mark(X)) [1] mark(nil) -> nil [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(s(X)) -> s(mark(X)) [1] mark(0) -> 0 [1] a__incr(X) -> incr(X) [1] a__adx(X) -> adx(X) [1] a__nats -> nats [1] a__zeros -> zeros [1] a__head(X) -> head(X) [1] a__tail(X) -> tail(X) [1] The TRS has the following type information: a__incr :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail nil :: nil:cons:s:incr:adx:0:zeros:nats:head:tail cons :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail s :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail mark :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail incr :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail a__adx :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail adx :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail a__nats :: nil:cons:s:incr:adx:0:zeros:nats:head:tail a__zeros :: nil:cons:s:incr:adx:0:zeros:nats:head:tail 0 :: nil:cons:s:incr:adx:0:zeros:nats:head:tail zeros :: nil:cons:s:incr:adx:0:zeros:nats:head:tail a__head :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail a__tail :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail nats :: nil:cons:s:incr:adx:0:zeros:nats:head:tail head :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail tail :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail Rewrite Strategy: INNERMOST ---------------------------------------- (249) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 2 0 => 0 zeros => 3 nats => 1 ---------------------------------------- (250) Obligation: Complexity RNTS consisting of the following rules: a__adx(z) -{ 1 }-> a__incr(1 + mark(X) + (1 + L)) :|: z = 1 + X + L, X >= 0, L >= 0 a__adx(z) -{ 1 }-> 2 :|: z = 2 a__adx(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__head(z) -{ 1 }-> mark(X) :|: z = 1 + X + L, X >= 0, L >= 0 a__head(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__incr(z) -{ 1 }-> 2 :|: z = 2 a__incr(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__incr(z) -{ 1 }-> 1 + (1 + mark(X)) + (1 + L) :|: z = 1 + X + L, X >= 0, L >= 0 a__nats -{ 1 }-> a__adx(a__zeros) :|: a__nats -{ 1 }-> 1 :|: a__tail(z) -{ 1 }-> mark(L) :|: z = 1 + X + L, X >= 0, L >= 0 a__tail(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__tail(mark(X)) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__nats :|: z = 1 mark(z) -{ 1 }-> a__incr(mark(X)) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__head(mark(X)) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__adx(mark(X)) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> 2 :|: z = 2 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. ---------------------------------------- (251) 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__zeros a__tail_1 a__adx_1 a__nats a__head_1 a__incr_1 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (252) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a__incr(nil) -> nil [1] a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L)) [1] a__adx(nil) -> nil [1] a__adx(cons(X, L)) -> a__incr(cons(mark(X), adx(L))) [1] a__nats -> a__adx(a__zeros) [1] a__zeros -> cons(0, zeros) [1] a__head(cons(X, L)) -> mark(X) [1] a__tail(cons(X, L)) -> mark(L) [1] mark(incr(X)) -> a__incr(mark(X)) [1] mark(adx(X)) -> a__adx(mark(X)) [1] mark(nats) -> a__nats [1] mark(zeros) -> a__zeros [1] mark(head(X)) -> a__head(mark(X)) [1] mark(tail(X)) -> a__tail(mark(X)) [1] mark(nil) -> nil [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(s(X)) -> s(mark(X)) [1] mark(0) -> 0 [1] a__incr(X) -> incr(X) [1] a__adx(X) -> adx(X) [1] a__nats -> nats [1] a__zeros -> zeros [1] a__head(X) -> head(X) [1] a__tail(X) -> tail(X) [1] The TRS has the following type information: a__incr :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail nil :: nil:cons:s:incr:adx:0:zeros:nats:head:tail cons :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail s :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail mark :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail incr :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail a__adx :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail adx :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail a__nats :: nil:cons:s:incr:adx:0:zeros:nats:head:tail a__zeros :: nil:cons:s:incr:adx:0:zeros:nats:head:tail 0 :: nil:cons:s:incr:adx:0:zeros:nats:head:tail zeros :: nil:cons:s:incr:adx:0:zeros:nats:head:tail a__head :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail a__tail :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail nats :: nil:cons:s:incr:adx:0:zeros:nats:head:tail head :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail tail :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail Rewrite Strategy: INNERMOST ---------------------------------------- (253) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (254) 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__incr(nil) -> nil [1] a__incr(cons(X, L)) -> cons(s(mark(X)), incr(L)) [1] a__adx(nil) -> nil [1] a__adx(cons(incr(X'), L)) -> a__incr(cons(a__incr(mark(X')), adx(L))) [2] a__adx(cons(adx(X''), L)) -> a__incr(cons(a__adx(mark(X'')), adx(L))) [2] a__adx(cons(nats, L)) -> a__incr(cons(a__nats, adx(L))) [2] a__adx(cons(zeros, L)) -> a__incr(cons(a__zeros, adx(L))) [2] a__adx(cons(head(X3), L)) -> a__incr(cons(a__head(mark(X3)), adx(L))) [2] a__adx(cons(tail(X4), L)) -> a__incr(cons(a__tail(mark(X4)), adx(L))) [2] a__adx(cons(nil, L)) -> a__incr(cons(nil, adx(L))) [2] a__adx(cons(cons(X1', X2'), L)) -> a__incr(cons(cons(mark(X1'), X2'), adx(L))) [2] a__adx(cons(s(X5), L)) -> a__incr(cons(s(mark(X5)), adx(L))) [2] a__adx(cons(0, L)) -> a__incr(cons(0, adx(L))) [2] a__nats -> a__adx(cons(0, zeros)) [2] a__nats -> a__adx(zeros) [2] a__zeros -> cons(0, zeros) [1] a__head(cons(X, L)) -> mark(X) [1] a__tail(cons(X, L)) -> mark(L) [1] mark(incr(incr(X6))) -> a__incr(a__incr(mark(X6))) [2] mark(incr(adx(X7))) -> a__incr(a__adx(mark(X7))) [2] mark(incr(nats)) -> a__incr(a__nats) [2] mark(incr(zeros)) -> a__incr(a__zeros) [2] mark(incr(head(X8))) -> a__incr(a__head(mark(X8))) [2] mark(incr(tail(X9))) -> a__incr(a__tail(mark(X9))) [2] mark(incr(nil)) -> a__incr(nil) [2] mark(incr(cons(X1'', X2''))) -> a__incr(cons(mark(X1''), X2'')) [2] mark(incr(s(X10))) -> a__incr(s(mark(X10))) [2] mark(incr(0)) -> a__incr(0) [2] mark(adx(incr(X11))) -> a__adx(a__incr(mark(X11))) [2] mark(adx(adx(X12))) -> a__adx(a__adx(mark(X12))) [2] mark(adx(nats)) -> a__adx(a__nats) [2] mark(adx(zeros)) -> a__adx(a__zeros) [2] mark(adx(head(X13))) -> a__adx(a__head(mark(X13))) [2] mark(adx(tail(X14))) -> a__adx(a__tail(mark(X14))) [2] mark(adx(nil)) -> a__adx(nil) [2] mark(adx(cons(X15, X21))) -> a__adx(cons(mark(X15), X21)) [2] mark(adx(s(X16))) -> a__adx(s(mark(X16))) [2] mark(adx(0)) -> a__adx(0) [2] mark(nats) -> a__nats [1] mark(zeros) -> a__zeros [1] mark(head(incr(X17))) -> a__head(a__incr(mark(X17))) [2] mark(head(adx(X18))) -> a__head(a__adx(mark(X18))) [2] mark(head(nats)) -> a__head(a__nats) [2] mark(head(zeros)) -> a__head(a__zeros) [2] mark(head(head(X19))) -> a__head(a__head(mark(X19))) [2] mark(head(tail(X20))) -> a__head(a__tail(mark(X20))) [2] mark(head(nil)) -> a__head(nil) [2] mark(head(cons(X110, X22))) -> a__head(cons(mark(X110), X22)) [2] mark(head(s(X23))) -> a__head(s(mark(X23))) [2] mark(head(0)) -> a__head(0) [2] mark(tail(incr(X24))) -> a__tail(a__incr(mark(X24))) [2] mark(tail(adx(X25))) -> a__tail(a__adx(mark(X25))) [2] mark(tail(nats)) -> a__tail(a__nats) [2] mark(tail(zeros)) -> a__tail(a__zeros) [2] mark(tail(head(X26))) -> a__tail(a__head(mark(X26))) [2] mark(tail(tail(X27))) -> a__tail(a__tail(mark(X27))) [2] mark(tail(nil)) -> a__tail(nil) [2] mark(tail(cons(X111, X28))) -> a__tail(cons(mark(X111), X28)) [2] mark(tail(s(X29))) -> a__tail(s(mark(X29))) [2] mark(tail(0)) -> a__tail(0) [2] mark(nil) -> nil [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(s(X)) -> s(mark(X)) [1] mark(0) -> 0 [1] a__incr(X) -> incr(X) [1] a__adx(X) -> adx(X) [1] a__nats -> nats [1] a__zeros -> zeros [1] a__head(X) -> head(X) [1] a__tail(X) -> tail(X) [1] The TRS has the following type information: a__incr :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail nil :: nil:cons:s:incr:adx:0:zeros:nats:head:tail cons :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail s :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail mark :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail incr :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail a__adx :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail adx :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail a__nats :: nil:cons:s:incr:adx:0:zeros:nats:head:tail a__zeros :: nil:cons:s:incr:adx:0:zeros:nats:head:tail 0 :: nil:cons:s:incr:adx:0:zeros:nats:head:tail zeros :: nil:cons:s:incr:adx:0:zeros:nats:head:tail a__head :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail a__tail :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail nats :: nil:cons:s:incr:adx:0:zeros:nats:head:tail head :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail tail :: nil:cons:s:incr:adx:0:zeros:nats:head:tail -> nil:cons:s:incr:adx:0:zeros:nats:head:tail Rewrite Strategy: INNERMOST ---------------------------------------- (255) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 2 0 => 0 zeros => 3 nats => 1 ---------------------------------------- (256) Obligation: Complexity RNTS consisting of the following rules: a__adx(z) -{ 2 }-> a__incr(1 + a__zeros + (1 + L)) :|: L >= 0, z = 1 + 3 + L a__adx(z) -{ 2 }-> a__incr(1 + a__tail(mark(X4)) + (1 + L)) :|: L >= 0, X4 >= 0, z = 1 + (1 + X4) + L a__adx(z) -{ 2 }-> a__incr(1 + a__nats + (1 + L)) :|: L >= 0, z = 1 + 1 + L a__adx(z) -{ 2 }-> a__incr(1 + a__incr(mark(X')) + (1 + L)) :|: X' >= 0, L >= 0, z = 1 + (1 + X') + L a__adx(z) -{ 2 }-> a__incr(1 + a__head(mark(X3)) + (1 + L)) :|: z = 1 + (1 + X3) + L, X3 >= 0, L >= 0 a__adx(z) -{ 2 }-> a__incr(1 + a__adx(mark(X'')) + (1 + L)) :|: z = 1 + (1 + X'') + L, X'' >= 0, L >= 0 a__adx(z) -{ 2 }-> a__incr(1 + 2 + (1 + L)) :|: z = 1 + 2 + L, L >= 0 a__adx(z) -{ 2 }-> a__incr(1 + 0 + (1 + L)) :|: z = 1 + 0 + L, L >= 0 a__adx(z) -{ 2 }-> a__incr(1 + (1 + mark(X5)) + (1 + L)) :|: z = 1 + (1 + X5) + L, X5 >= 0, L >= 0 a__adx(z) -{ 2 }-> a__incr(1 + (1 + mark(X1') + X2') + (1 + L)) :|: X2' >= 0, X1' >= 0, L >= 0, z = 1 + (1 + X1' + X2') + L a__adx(z) -{ 1 }-> 2 :|: z = 2 a__adx(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__head(z) -{ 1 }-> mark(X) :|: z = 1 + X + L, X >= 0, L >= 0 a__head(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__incr(z) -{ 1 }-> 2 :|: z = 2 a__incr(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__incr(z) -{ 1 }-> 1 + (1 + mark(X)) + (1 + L) :|: z = 1 + X + L, X >= 0, L >= 0 a__nats -{ 2 }-> a__adx(3) :|: a__nats -{ 2 }-> a__adx(1 + 0 + 3) :|: a__nats -{ 1 }-> 1 :|: a__tail(z) -{ 1 }-> mark(L) :|: z = 1 + X + L, X >= 0, L >= 0 a__tail(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__tail(a__zeros) :|: z = 1 + 3 mark(z) -{ 2 }-> a__tail(a__tail(mark(X27))) :|: X27 >= 0, z = 1 + (1 + X27) mark(z) -{ 2 }-> a__tail(a__nats) :|: z = 1 + 1 mark(z) -{ 2 }-> a__tail(a__incr(mark(X24))) :|: z = 1 + (1 + X24), X24 >= 0 mark(z) -{ 2 }-> a__tail(a__head(mark(X26))) :|: X26 >= 0, z = 1 + (1 + X26) mark(z) -{ 2 }-> a__tail(a__adx(mark(X25))) :|: z = 1 + (1 + X25), X25 >= 0 mark(z) -{ 2 }-> a__tail(2) :|: z = 1 + 2 mark(z) -{ 2 }-> a__tail(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__tail(1 + mark(X29)) :|: X29 >= 0, z = 1 + (1 + X29) mark(z) -{ 2 }-> a__tail(1 + mark(X111) + X28) :|: X28 >= 0, z = 1 + (1 + X111 + X28), X111 >= 0 mark(z) -{ 1 }-> a__nats :|: z = 1 mark(z) -{ 2 }-> a__incr(a__zeros) :|: z = 1 + 3 mark(z) -{ 2 }-> a__incr(a__tail(mark(X9))) :|: X9 >= 0, z = 1 + (1 + X9) mark(z) -{ 2 }-> a__incr(a__nats) :|: z = 1 + 1 mark(z) -{ 2 }-> a__incr(a__incr(mark(X6))) :|: X6 >= 0, z = 1 + (1 + X6) mark(z) -{ 2 }-> a__incr(a__head(mark(X8))) :|: X8 >= 0, z = 1 + (1 + X8) mark(z) -{ 2 }-> a__incr(a__adx(mark(X7))) :|: X7 >= 0, z = 1 + (1 + X7) mark(z) -{ 2 }-> a__incr(2) :|: z = 1 + 2 mark(z) -{ 2 }-> a__incr(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__incr(1 + mark(X10)) :|: z = 1 + (1 + X10), X10 >= 0 mark(z) -{ 2 }-> a__incr(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0 mark(z) -{ 2 }-> a__head(a__zeros) :|: z = 1 + 3 mark(z) -{ 2 }-> a__head(a__tail(mark(X20))) :|: z = 1 + (1 + X20), X20 >= 0 mark(z) -{ 2 }-> a__head(a__nats) :|: z = 1 + 1 mark(z) -{ 2 }-> a__head(a__incr(mark(X17))) :|: X17 >= 0, z = 1 + (1 + X17) mark(z) -{ 2 }-> a__head(a__head(mark(X19))) :|: X19 >= 0, z = 1 + (1 + X19) mark(z) -{ 2 }-> a__head(a__adx(mark(X18))) :|: X18 >= 0, z = 1 + (1 + X18) mark(z) -{ 2 }-> a__head(2) :|: z = 1 + 2 mark(z) -{ 2 }-> a__head(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__head(1 + mark(X23)) :|: z = 1 + (1 + X23), X23 >= 0 mark(z) -{ 2 }-> a__head(1 + mark(X110) + X22) :|: X110 >= 0, X22 >= 0, z = 1 + (1 + X110 + X22) mark(z) -{ 2 }-> a__adx(a__zeros) :|: z = 1 + 3 mark(z) -{ 2 }-> a__adx(a__tail(mark(X14))) :|: z = 1 + (1 + X14), X14 >= 0 mark(z) -{ 2 }-> a__adx(a__nats) :|: z = 1 + 1 mark(z) -{ 2 }-> a__adx(a__incr(mark(X11))) :|: X11 >= 0, z = 1 + (1 + X11) mark(z) -{ 2 }-> a__adx(a__head(mark(X13))) :|: z = 1 + (1 + X13), X13 >= 0 mark(z) -{ 2 }-> a__adx(a__adx(mark(X12))) :|: z = 1 + (1 + X12), X12 >= 0 mark(z) -{ 2 }-> a__adx(2) :|: z = 1 + 2 mark(z) -{ 2 }-> a__adx(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__adx(1 + mark(X16)) :|: X16 >= 0, z = 1 + (1 + X16) mark(z) -{ 2 }-> a__adx(1 + mark(X15) + X21) :|: X21 >= 0, z = 1 + (1 + X15 + X21), X15 >= 0 mark(z) -{ 1 }-> 2 :|: z = 2 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 ---------------------------------------- (257) 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 :|: ---------------------------------------- (258) Obligation: Complexity RNTS consisting of the following rules: a__adx(z) -{ 2 }-> a__incr(1 + a__tail(mark(X4)) + (1 + L)) :|: L >= 0, X4 >= 0, z = 1 + (1 + X4) + L a__adx(z) -{ 2 }-> a__incr(1 + a__nats + (1 + L)) :|: L >= 0, z = 1 + 1 + L a__adx(z) -{ 2 }-> a__incr(1 + a__incr(mark(X')) + (1 + L)) :|: X' >= 0, L >= 0, z = 1 + (1 + X') + L a__adx(z) -{ 2 }-> a__incr(1 + a__head(mark(X3)) + (1 + L)) :|: z = 1 + (1 + X3) + L, X3 >= 0, L >= 0 a__adx(z) -{ 2 }-> a__incr(1 + a__adx(mark(X'')) + (1 + L)) :|: z = 1 + (1 + X'') + L, X'' >= 0, L >= 0 a__adx(z) -{ 3 }-> a__incr(1 + 3 + (1 + L)) :|: L >= 0, z = 1 + 3 + L a__adx(z) -{ 2 }-> a__incr(1 + 2 + (1 + L)) :|: z = 1 + 2 + L, L >= 0 a__adx(z) -{ 2 }-> a__incr(1 + 0 + (1 + L)) :|: z = 1 + 0 + L, L >= 0 a__adx(z) -{ 2 }-> a__incr(1 + (1 + mark(X5)) + (1 + L)) :|: z = 1 + (1 + X5) + L, X5 >= 0, L >= 0 a__adx(z) -{ 2 }-> a__incr(1 + (1 + mark(X1') + X2') + (1 + L)) :|: X2' >= 0, X1' >= 0, L >= 0, z = 1 + (1 + X1' + X2') + L a__adx(z) -{ 3 }-> a__incr(1 + (1 + 0 + 3) + (1 + L)) :|: L >= 0, z = 1 + 3 + L a__adx(z) -{ 1 }-> 2 :|: z = 2 a__adx(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__head(z) -{ 1 }-> mark(X) :|: z = 1 + X + L, X >= 0, L >= 0 a__head(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__incr(z) -{ 1 }-> 2 :|: z = 2 a__incr(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__incr(z) -{ 1 }-> 1 + (1 + mark(X)) + (1 + L) :|: z = 1 + X + L, X >= 0, L >= 0 a__nats -{ 2 }-> a__adx(3) :|: a__nats -{ 2 }-> a__adx(1 + 0 + 3) :|: a__nats -{ 1 }-> 1 :|: a__tail(z) -{ 1 }-> mark(L) :|: z = 1 + X + L, X >= 0, L >= 0 a__tail(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__zeros -{ 1 }-> 3 :|: a__zeros -{ 1 }-> 1 + 0 + 3 :|: mark(z) -{ 2 }-> a__tail(a__tail(mark(X27))) :|: X27 >= 0, z = 1 + (1 + X27) mark(z) -{ 2 }-> a__tail(a__nats) :|: z = 1 + 1 mark(z) -{ 2 }-> a__tail(a__incr(mark(X24))) :|: z = 1 + (1 + X24), X24 >= 0 mark(z) -{ 2 }-> a__tail(a__head(mark(X26))) :|: X26 >= 0, z = 1 + (1 + X26) mark(z) -{ 2 }-> a__tail(a__adx(mark(X25))) :|: z = 1 + (1 + X25), X25 >= 0 mark(z) -{ 3 }-> a__tail(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__tail(2) :|: z = 1 + 2 mark(z) -{ 2 }-> a__tail(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__tail(1 + mark(X29)) :|: X29 >= 0, z = 1 + (1 + X29) mark(z) -{ 2 }-> a__tail(1 + mark(X111) + X28) :|: X28 >= 0, z = 1 + (1 + X111 + X28), X111 >= 0 mark(z) -{ 3 }-> a__tail(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 1 }-> a__nats :|: z = 1 mark(z) -{ 2 }-> a__incr(a__tail(mark(X9))) :|: X9 >= 0, z = 1 + (1 + X9) mark(z) -{ 2 }-> a__incr(a__nats) :|: z = 1 + 1 mark(z) -{ 2 }-> a__incr(a__incr(mark(X6))) :|: X6 >= 0, z = 1 + (1 + X6) mark(z) -{ 2 }-> a__incr(a__head(mark(X8))) :|: X8 >= 0, z = 1 + (1 + X8) mark(z) -{ 2 }-> a__incr(a__adx(mark(X7))) :|: X7 >= 0, z = 1 + (1 + X7) mark(z) -{ 3 }-> a__incr(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__incr(2) :|: z = 1 + 2 mark(z) -{ 2 }-> a__incr(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__incr(1 + mark(X10)) :|: z = 1 + (1 + X10), X10 >= 0 mark(z) -{ 2 }-> a__incr(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0 mark(z) -{ 3 }-> a__incr(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__head(a__tail(mark(X20))) :|: z = 1 + (1 + X20), X20 >= 0 mark(z) -{ 2 }-> a__head(a__nats) :|: z = 1 + 1 mark(z) -{ 2 }-> a__head(a__incr(mark(X17))) :|: X17 >= 0, z = 1 + (1 + X17) mark(z) -{ 2 }-> a__head(a__head(mark(X19))) :|: X19 >= 0, z = 1 + (1 + X19) mark(z) -{ 2 }-> a__head(a__adx(mark(X18))) :|: X18 >= 0, z = 1 + (1 + X18) mark(z) -{ 3 }-> a__head(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__head(2) :|: z = 1 + 2 mark(z) -{ 2 }-> a__head(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__head(1 + mark(X23)) :|: z = 1 + (1 + X23), X23 >= 0 mark(z) -{ 2 }-> a__head(1 + mark(X110) + X22) :|: X110 >= 0, X22 >= 0, z = 1 + (1 + X110 + X22) mark(z) -{ 3 }-> a__head(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__adx(a__tail(mark(X14))) :|: z = 1 + (1 + X14), X14 >= 0 mark(z) -{ 2 }-> a__adx(a__nats) :|: z = 1 + 1 mark(z) -{ 2 }-> a__adx(a__incr(mark(X11))) :|: X11 >= 0, z = 1 + (1 + X11) mark(z) -{ 2 }-> a__adx(a__head(mark(X13))) :|: z = 1 + (1 + X13), X13 >= 0 mark(z) -{ 2 }-> a__adx(a__adx(mark(X12))) :|: z = 1 + (1 + X12), X12 >= 0 mark(z) -{ 3 }-> a__adx(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__adx(2) :|: z = 1 + 2 mark(z) -{ 2 }-> a__adx(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__adx(1 + mark(X16)) :|: X16 >= 0, z = 1 + (1 + X16) mark(z) -{ 2 }-> a__adx(1 + mark(X15) + X21) :|: X21 >= 0, z = 1 + (1 + X15 + X21), X15 >= 0 mark(z) -{ 3 }-> a__adx(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 2 }-> 3 :|: z = 3 mark(z) -{ 1 }-> 2 :|: z = 2 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 ---------------------------------------- (259) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (260) Obligation: Complexity RNTS consisting of the following rules: a__adx(z) -{ 2 }-> a__incr(1 + a__tail(mark(X4)) + (1 + L)) :|: L >= 0, X4 >= 0, z = 1 + (1 + X4) + L a__adx(z) -{ 2 }-> a__incr(1 + a__nats + (1 + (z - 2))) :|: z - 2 >= 0 a__adx(z) -{ 2 }-> a__incr(1 + a__incr(mark(X')) + (1 + L)) :|: X' >= 0, L >= 0, z = 1 + (1 + X') + L a__adx(z) -{ 2 }-> a__incr(1 + a__head(mark(X3)) + (1 + L)) :|: z = 1 + (1 + X3) + L, X3 >= 0, L >= 0 a__adx(z) -{ 2 }-> a__incr(1 + a__adx(mark(X'')) + (1 + L)) :|: z = 1 + (1 + X'') + L, X'' >= 0, L >= 0 a__adx(z) -{ 3 }-> a__incr(1 + 3 + (1 + (z - 4))) :|: z - 4 >= 0 a__adx(z) -{ 2 }-> a__incr(1 + 2 + (1 + (z - 3))) :|: z - 3 >= 0 a__adx(z) -{ 2 }-> a__incr(1 + 0 + (1 + (z - 1))) :|: z - 1 >= 0 a__adx(z) -{ 2 }-> a__incr(1 + (1 + mark(X5)) + (1 + L)) :|: z = 1 + (1 + X5) + L, X5 >= 0, L >= 0 a__adx(z) -{ 2 }-> a__incr(1 + (1 + mark(X1') + X2') + (1 + L)) :|: X2' >= 0, X1' >= 0, L >= 0, z = 1 + (1 + X1' + X2') + L a__adx(z) -{ 3 }-> a__incr(1 + (1 + 0 + 3) + (1 + (z - 4))) :|: z - 4 >= 0 a__adx(z) -{ 1 }-> 2 :|: z = 2 a__adx(z) -{ 1 }-> 1 + z :|: z >= 0 a__head(z) -{ 1 }-> mark(X) :|: z = 1 + X + L, X >= 0, L >= 0 a__head(z) -{ 1 }-> 1 + z :|: z >= 0 a__incr(z) -{ 1 }-> 2 :|: z = 2 a__incr(z) -{ 1 }-> 1 + z :|: z >= 0 a__incr(z) -{ 1 }-> 1 + (1 + mark(X)) + (1 + L) :|: z = 1 + X + L, X >= 0, L >= 0 a__nats -{ 2 }-> a__adx(3) :|: a__nats -{ 2 }-> a__adx(1 + 0 + 3) :|: a__nats -{ 1 }-> 1 :|: a__tail(z) -{ 1 }-> mark(L) :|: z = 1 + X + L, X >= 0, L >= 0 a__tail(z) -{ 1 }-> 1 + z :|: z >= 0 a__zeros -{ 1 }-> 3 :|: a__zeros -{ 1 }-> 1 + 0 + 3 :|: mark(z) -{ 2 }-> a__tail(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__tail(a__nats) :|: z = 1 + 1 mark(z) -{ 2 }-> a__tail(a__incr(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__tail(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__tail(a__adx(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 3 }-> a__tail(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__tail(2) :|: z = 1 + 2 mark(z) -{ 2 }-> a__tail(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__tail(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__tail(1 + mark(X111) + X28) :|: X28 >= 0, z = 1 + (1 + X111 + X28), X111 >= 0 mark(z) -{ 3 }-> a__tail(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 1 }-> a__nats :|: z = 1 mark(z) -{ 2 }-> a__incr(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__incr(a__nats) :|: z = 1 + 1 mark(z) -{ 2 }-> a__incr(a__incr(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__incr(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__incr(a__adx(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 3 }-> a__incr(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__incr(2) :|: z = 1 + 2 mark(z) -{ 2 }-> a__incr(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__incr(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__incr(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0 mark(z) -{ 3 }-> a__incr(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__head(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__head(a__nats) :|: z = 1 + 1 mark(z) -{ 2 }-> a__head(a__incr(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__head(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__head(a__adx(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 3 }-> a__head(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__head(2) :|: z = 1 + 2 mark(z) -{ 2 }-> a__head(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__head(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__head(1 + mark(X110) + X22) :|: X110 >= 0, X22 >= 0, z = 1 + (1 + X110 + X22) mark(z) -{ 3 }-> a__head(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__adx(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__adx(a__nats) :|: z = 1 + 1 mark(z) -{ 2 }-> a__adx(a__incr(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__adx(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__adx(a__adx(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 3 }-> a__adx(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__adx(2) :|: z = 1 + 2 mark(z) -{ 2 }-> a__adx(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__adx(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__adx(1 + mark(X15) + X21) :|: X21 >= 0, z = 1 + (1 + X15 + X21), X15 >= 0 mark(z) -{ 3 }-> a__adx(1 + 0 + 3) :|: z = 1 + 3 mark(z) -{ 2 }-> 3 :|: z = 3 mark(z) -{ 1 }-> 2 :|: z = 2 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