KILLED proof of input_UxfhSyr5e6.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 9 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) InliningProof [UPPER BOUND(ID), 283 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CompletionProof [UPPER BOUND(ID), 0 ms] (20) CpxTypedWeightedCompleteTrs (21) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms] (22) CpxRNTS (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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (32) CdtProblem (33) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 10 ms] (36) CdtProblem (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms] (48) CdtProblem (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 6 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 9 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (110) CdtProblem (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 1 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 4 ms] (144) CdtProblem (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 47 ms] (154) CdtProblem (155) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 61 ms] (158) CdtProblem (159) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 148 ms] (162) CdtProblem (163) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 155 ms] (166) CdtProblem (167) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 273 ms] (170) CdtProblem (171) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 247 ms] (174) CdtProblem (175) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 329 ms] (178) CdtProblem (179) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 346 ms] (182) CdtProblem (183) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 352 ms] (186) CdtProblem (187) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 385 ms] (190) CdtProblem (191) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 418 ms] (194) CdtProblem (195) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 471 ms] (198) CdtProblem (199) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 551 ms] (202) CdtProblem (203) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 517 ms] (206) CdtProblem (207) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 568 ms] (210) CdtProblem (211) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 627 ms] (214) CdtProblem (215) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 707 ms] (218) CdtProblem (219) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (220) CdtProblem (221) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 685 ms] (222) CdtProblem (223) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a__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) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a__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 ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a__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 ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: mark_1 a__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 ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a__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 ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a__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 ---------------------------------------- (13) 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 ---------------------------------------- (14) 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 ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: a__zeros -{ 1 }-> 3 :|: a__zeros -{ 1 }-> 1 + 0 + 3 :|: ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: a__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 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: a__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 ---------------------------------------- (19) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (20) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a__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 ---------------------------------------- (21) 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 ---------------------------------------- (22) 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. ---------------------------------------- (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(z0) -> c5 A__TAIL(z0) -> c13 A__HEAD(z0) -> c11 A__NATS -> c7 A__INCR(z0) -> c2 MARK(0) -> c23 A__ZEROS -> c9 A__INCR(nil) -> c MARK(nil) -> c20 MARK(zeros) -> c17(A__ZEROS) A__ADX(nil) -> c3 A__ZEROS -> c8 ---------------------------------------- (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) 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)) ---------------------------------------- (30) 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 ---------------------------------------- (31) 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)) ---------------------------------------- (32) 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 ---------------------------------------- (33) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (34) 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 ---------------------------------------- (35) 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)) ---------------------------------------- (36) 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 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(adx(0)) -> c15(A__ADX(0), MARK(0)) MARK(adx(nil)) -> c15(A__ADX(nil), MARK(nil)) ---------------------------------------- (38) 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 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (40) 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 ---------------------------------------- (41) 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)) ---------------------------------------- (42) 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 ---------------------------------------- (43) 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)) ---------------------------------------- (44) 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 ---------------------------------------- (45) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (46) 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 ---------------------------------------- (47) 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)) ---------------------------------------- (48) 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 ---------------------------------------- (49) 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)) ---------------------------------------- (50) 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 ---------------------------------------- (51) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (53) 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)) ---------------------------------------- (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(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 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__NATS -> c6(A__ADX(zeros)) ---------------------------------------- (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(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 ---------------------------------------- (57) 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))) ---------------------------------------- (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(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 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (61) 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))) ---------------------------------------- (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(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 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (65) 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)) ---------------------------------------- (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(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 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 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(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 ---------------------------------------- (69) 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))) ---------------------------------------- (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)) 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 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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)) 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 ---------------------------------------- (73) 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))) ---------------------------------------- (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)) 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 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (77) 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)) ---------------------------------------- (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(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 ---------------------------------------- (79) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(zeros)) -> c14(A__INCR(zeros)) ---------------------------------------- (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(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 ---------------------------------------- (81) 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))) ---------------------------------------- (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(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 ---------------------------------------- (83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (85) 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))) ---------------------------------------- (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(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 ---------------------------------------- (87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (89) 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)) ---------------------------------------- (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(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 ---------------------------------------- (91) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (93) 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))) ---------------------------------------- (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(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 ---------------------------------------- (95) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (97) 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))) ---------------------------------------- (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(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 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (101) 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)) ---------------------------------------- (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(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 ---------------------------------------- (103) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(adx(zeros)) -> c15(A__ADX(zeros)) ---------------------------------------- (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(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 ---------------------------------------- (105) 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))) ---------------------------------------- (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(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 ---------------------------------------- (107) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (109) 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))) ---------------------------------------- (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(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 ---------------------------------------- (111) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (113) 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)) ---------------------------------------- (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(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 ---------------------------------------- (115) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(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))) -> 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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 ---------------------------------------- (117) 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))) ---------------------------------------- (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(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 ---------------------------------------- (119) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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)) 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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))) -> 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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 ---------------------------------------- (121) 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))) ---------------------------------------- (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(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)))) -> 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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 ---------------------------------------- (123) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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)))) 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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 ---------------------------------------- (125) 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)) ---------------------------------------- (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(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 ---------------------------------------- (127) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(head(zeros)) -> c18(A__HEAD(zeros)) ---------------------------------------- (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(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)))), 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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)))) -> 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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))), 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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 ---------------------------------------- (129) 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))) ---------------------------------------- (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(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)))), 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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)))) 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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 ---------------------------------------- (131) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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))) -> 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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 ---------------------------------------- (133) 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))) ---------------------------------------- (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(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)))), 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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)))) -> 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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 ---------------------------------------- (135) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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)), 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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 ---------------------------------------- (137) 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)) ---------------------------------------- (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(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)))) 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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 ---------------------------------------- (139) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(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)))), 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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 ---------------------------------------- (141) 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))) ---------------------------------------- (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(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)))) 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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))) 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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 ---------------------------------------- (143) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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)))) 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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 ---------------------------------------- (145) 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))) ---------------------------------------- (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(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))) -> 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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)) 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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))) -> 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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 ---------------------------------------- (147) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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))) 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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 ---------------------------------------- (149) 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)) ---------------------------------------- (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(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), 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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 ---------------------------------------- (151) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(tail(zeros)) -> c19(A__TAIL(zeros)) ---------------------------------------- (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(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, 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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))) 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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 ---------------------------------------- (153) 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))) ---------------------------------------- (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) -> 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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 ---------------------------------------- (155) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(incr(incr(nil))) -> c14(A__INCR(incr(nil))) MARK(incr(incr(nil))) -> c14(A__INCR(nil)) ---------------------------------------- (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(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)))) -> 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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 ---------------------------------------- (157) 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))) ---------------------------------------- (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(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)))) 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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)))) -> 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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 ---------------------------------------- (159) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(incr(0))) -> c14(A__INCR(incr(0))) ---------------------------------------- (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(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)))) -> 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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 ---------------------------------------- (161) 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))) ---------------------------------------- (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(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)))) 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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))) -> 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-> 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)) -> 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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)))) -> 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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 ---------------------------------------- (163) 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)) ---------------------------------------- (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(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))) -> 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-> 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)) 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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 ---------------------------------------- (165) 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))) ---------------------------------------- (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(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)))) -> 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-> 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 ---------------------------------------- (167) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(adx(0))) -> c14(A__INCR(adx(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(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)))) -> 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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 ---------------------------------------- (169) 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))) ---------------------------------------- (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(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)))) -> 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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 ---------------------------------------- (171) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(head(nil))) -> c14(A__INCR(head(nil))) ---------------------------------------- (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(adx(incr(z0)))) -> c14(A__INCR(a__adx(a__incr(mark(z0)))), MARK(adx(incr(z0)))) MARK(incr(adx(adx(z0)))) -> 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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)))) -> 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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 ---------------------------------------- (173) 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))) ---------------------------------------- (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(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)))) -> 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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 ---------------------------------------- (175) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(head(0))) -> c14(A__INCR(head(0))) ---------------------------------------- (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(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))) 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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 ---------------------------------------- (177) 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))) ---------------------------------------- (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(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)))) -> 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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 ---------------------------------------- (179) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(tail(nil))) -> c14(A__INCR(tail(nil))) ---------------------------------------- (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))) -> 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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)))), 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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 ---------------------------------------- (181) 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))) ---------------------------------------- (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(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)), 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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))) 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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)))), 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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 ---------------------------------------- (183) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(tail(0))) -> c14(A__INCR(tail(0))) ---------------------------------------- (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) -> 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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 ---------------------------------------- (185) 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))) ---------------------------------------- (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(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))) 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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 ---------------------------------------- (187) 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))) ---------------------------------------- (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(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)))) 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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 ---------------------------------------- (189) 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))) ---------------------------------------- (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(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))), 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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 ---------------------------------------- (191) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(adx(incr(0))) -> c15(A__ADX(incr(0))) ---------------------------------------- (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(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)))) -> 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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 ---------------------------------------- (193) 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))) ---------------------------------------- (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(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 ---------------------------------------- (195) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(adx(adx(nil))) -> c15(A__ADX(nil)) MARK(adx(adx(nil))) -> c15(A__ADX(adx(nil))) ---------------------------------------- (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))) 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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, 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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 ---------------------------------------- (197) 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))) ---------------------------------------- (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(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))) -> 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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 ---------------------------------------- (199) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(adx(adx(0))) -> c15(A__ADX(adx(0))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> 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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, 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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))) 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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)))) -> 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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 ---------------------------------------- (201) 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))) ---------------------------------------- (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(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))) 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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 ---------------------------------------- (203) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(adx(head(nil))) -> c15(A__ADX(head(nil))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: a__incr(nil) -> nil a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) 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-> 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)))) -> 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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)))), 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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 ---------------------------------------- (205) 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))) ---------------------------------------- (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)))) -> 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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 ---------------------------------------- (207) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(adx(head(0))) -> c15(A__ADX(head(0))) ---------------------------------------- (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)))) 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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 ---------------------------------------- (209) 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))) ---------------------------------------- (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))) 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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 ---------------------------------------- (211) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(adx(tail(nil))) -> c15(A__ADX(tail(nil))) ---------------------------------------- (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(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, 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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)))) -> 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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)))), 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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)))) -> 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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 ---------------------------------------- (213) 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))) ---------------------------------------- (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(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))) 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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 ---------------------------------------- (215) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(adx(tail(0))) -> c15(A__ADX(tail(0))) ---------------------------------------- (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) 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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)))) 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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 ---------------------------------------- (217) 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))) ---------------------------------------- (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(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)), 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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)))), 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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 ---------------------------------------- (219) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(head(incr(nil))) -> c18(A__HEAD(nil)) MARK(head(incr(nil))) -> c18(A__HEAD(incr(nil))) ---------------------------------------- (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) -> 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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)))) 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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 ---------------------------------------- (221) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(incr(0))) -> c18(A__HEAD(a__incr(0))) by MARK(head(incr(0))) -> c18(A__HEAD(incr(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)), 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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(0))) -> c18(A__HEAD(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 ---------------------------------------- (223) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(head(incr(0))) -> c18(A__HEAD(incr(0))) ---------------------------------------- (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) -> 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-> 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)))) -> 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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